Would forcing the new citizens and prosecuting anyone that tried to spoke a different language be justified for the better end?
No. The End Sucks. So Do The Middle and Beginning.
It creates a privileged culture, suppresses others, leads to language stagnation, thought crime, and a police state.
This has been tried a number of times throughout history, most notably Turkey or the US forcing Native Americans and Australia forcing Aborigines to learn English. Usually it results in continuous cultural suppression. It establishes a dominant culture and suppresses the others.
An analogy is enforcing a dress code. The choices you make in what that code is and how its enforced says which culture and economic class gets preference, sets them at a higher standard of what is "proper", and selective enforcement can be used for harassment. For example, a "no hoodies" rule is clearly a way to target certain racial and economic classes, and deciding what is and is not a "hoodie" can be used to harass.
Here's the choices you need to make and how they lead to that end.
Which Language and Why?
I'm gonna use the US as an example to make this less abstract.
Which language do you choose? That might seem obvious, whatever language the most people already speak (English)... or do you go with one that's easy to learn (Spanish)? Maybe the pick the one with the most in common with other languages (... maybe Spanish again)? Or do you use the language that the most of your neighbors and trading partners speak (Mandarin)? Or do you make up a whole new language so everyone is equally hosed (Esperanto)?
Whatever you pick, it's going to be a continuing hassle.
Language As Privilege
If you chose an existing language, existing speakers are now privileged.
If you create a new language, well-off people are now privileged because they have the free time and money to learn the new language, hire tutors, training programs, etc...
Let's say it's English. Existing English speakers don't have to do anything. They don't have to spend time, and money, for schooling. They can keep their existing jobs, in fact native speakers will be in high demand as everyone else tries to catch up. The industry of teaching English will expand creating more jobs for English speakers.
As long as you are trying to speak it you are welcome.
This creates a linguistic, cultural, and economic privilege to immigration. You're welcome... so long as you either already speak the language, or have the money and free time to learn it.
Newcomers who are "trying" to speak the language remain at a disadvantage. Their native language is not just not spoken, it is illegal. How do they get a job? How do they read a contract? How do they read a manual?
This doesn't end.
Language As Culture
The language you speak isn't just some interchangeable part. It is your culture and it even alters how you think. It's your written and oral histories, parables, stories, songs, expressions, and vocabulary. All these things are made illegal. It wipes out other cultures.
Learning Material As Cultural Indoctrination
Until recently, the most translated book was the Bible. Missionaries were happy to teach you how to read... but it was going to be a Bible.
Now we're not quite as blunt about our indoctrination, but when you're running an entire society through a forced language re-education program the choice of reading material, pictures, phrasing, vocabulary, and grammar will be indoctrinate a certain world view whether you mean it or not.
Even something as innocent as your choice of noun to use when teaching basic grammar can codify what is normal and what is not.
I like to eat apples.
People eat apples.
Apples are good for you.
They bought three apples.
Translation As Suppression
Anyone who says "just translate them" has never done translation. Meaning is lost between languages, especially for songs. Translation is always a trade off between the literal meaning and the metaphorical meaning. How do you translate "stop, you're killing me"?
You can't translate everything, there simply aren't enough translators and money to pay them. The selection of what gets translated and how it gets translated becomes cultural suppression.
And if there aren't a lot of translators for your language... oh well. It disappears.
Translation As Rewriting Cultural History
Translators have a lot of power to subtly shape our understanding of history when the original documents are in another language. Not just in the choice of what gets translated, but the choices they make in doing that translation.
Perhaps the most famous example is the bowdlerization of classic texts such as Shakespeare, Ancient Greek and Roman writers, and the Bible. Every language has ambiguity through idioms and context.
Embarrassing historical documents can be subtly reworded to make them seem innocuous. Homosexuality, sex, dirty jokes... anything considered "vulgar" were obscured in popular translations of Greek and Roman texts to prop up the idea that this was a prim and proper golden age of humanity and that our morals have slipped. Have a look into the writings of Martial and Catullus or watch an uncensored version of Lysistrata.
For more on this read The Harlot by the Side of the Road: Forbidden Tales of the Bible which puts racy Bible stories in plain English and reveals what they're really talking about.
The End Is Bad.
The end doesn't justify the means because enforcing cultural laws leads you to bad outcomes.
Option 1: Surveillance State
How do you enforce the language law? How do you check that people aren't speaking and writing their own languages? To enforce this requires a surveillance state.
You need to spy on what people are saying, and what people are writing. This means no strong encryption. This means neighbors turning in neighbors. This means crackdowns on cultural displays like something as innocuous as showing a subtitled movie or displaying an old, untranslated play.
Option 2: Selective Enforcement As Cultural Domination
In this option, you count on normal law enforcement to enforce the language law. For otherwise innocuous, culturally driven laws like this there is a great urge to use selective enforcement as police harassment of groups they don't like. Want to harass someone? Accuse them of "speaking foreign".
Establishing One Language/Culture As "Better"
As with dress codes, choosing a single language signals that one language is "better" than the others. Again, doesn't matter what your intent is, people will use this as an excuse or grow up with this lesson.
Suppressing Knowledge Of Other Cultures
Since nobody is allowed to practice other languages, how can they ever really understand other cultures? People who only speak one language get a very selective and limited view of the world.
If you travel to a foreign country, you can only speak and read things which are in your language. Everything else is out of bounds, or you need a phrase book, or hire a translator (more economic privilege). This will keep most people to "tourist" areas and they will get a very selected and limited view.
This will twist your people's understanding of the world. Their limited view through their own lens will encourage xenophobia.
Language Police, Language Stagnation
The language can evolve as long as any changes to it would be applied globally.
Language evolves and changes. But in your world these changes have to first be approved before they can be legally used. Since they can't be used legally, the population can't first play around with them to see what works. Some council of Language Police decides what new words the people need (or, oh god, the people vote on what new words are ok?).
It's the ultimate in Linguistic prescription. At best this promotes cultural stagnation as the language is not allowed to naturally change and adapt. For example, as much as some people don't like it, verbing nouns is really useful.
At worst, constant tinkering with the language creates continual, punctuated, and awkward changes that everyone needs to relearn.
As an example, the speeches of Atatürk in the 1920s, leader of the Turkish language reforms, cannot be understood by most modern Turks.
What words, spellings, and phrases the Language Police decide are legal will be informed by what cultural, economic, and political things they are associated with. As a simple example, I was told growing up that "ain't" ain't a word, even though the meaning is clear, because it's considered lower class.
Another example is "sodomy". Want to suppress certain sexual acts through language? Roll them all into one word, now it's really easy to over-generalize, and difficult to discuss in detail. Then attach to them a word that says "remember that time God smote a whole city for being perverts?"
Similar examples come from dress codes. Not just things like hoodie bans, but here's a list of banned items from a bar using "safety" as cultural suppression.
Most of can be justified with safety, but others are simply targeting certain cultures they don't want around. Pacifiers, glowsticks, stuffed animals, and candy bracelets are stereotypical of raver culture. Others like "no chapstick" might be some sort of attempt to stop secret drug use?
Language restrictions can be used similarly.
What if there simply isn't a word to express what you want to say? Or an idea that you came up with? How many words for emotions and thoughts and actions have we come up with in the past ten years of ubiquitous Internet use and loan words alone? Mansplaining; lol; owned; email; to email; code monkey; texting; sexting; burner account; DOXing.... off the top of my head. Sure, there are similar words and phrases, but they don't say it quite the same way.
An example of vocabulary shaping thinking is the German loanword schadenfreude. We have sadism and masochism, a relationship of pain and pleasure between two parties... but schadenfreude is the relationship of a third and otherwise unrelated party. Sure, you can express this idea without introducing a new word, but a word neatly packages an idea for transmission to others.
Must we use increasingly awkward and literal phrases? Hey, are you going to the musical show which is a darker offshoot to the backlash to the cultural and economic stagnation represented by rock and roll tonight? A goth-punk show. Do we always have to point out that L.A.S.E.R. is an acronym for Light Amplification by Stimulated Emission of Radiation?
What about poetry and music? Will an artist be prosecuted for using a word in a non-approved way?
I could go on, but this is turning into a dissertation on cultural identity and suppression. I think you get the idea.