How do we make the impossible, possible? We begin by explaining why it's impossible.
WB sees a lot of questions of the form, "how do I stop X from ever happening?" X can be a technology, or an historical event, or a lot of other things. When it comes to a technology like tall buildings, there's a critical issue you really need to understand.
Technology happens. I know that sounds trite, but stopping technological development is whomping hard if not down right impossible. We're standing atop a pyramid of knowledge and experience and you're talking about taking a chunk out of the middle somewhere. It shouldn't surprise you that it will either (a) not have an effect or (b) prove impossible because all the other chunks around it are still there. In other words, you could say that steel wasn't invented or concrete wasn't invented or anything else, but in reality, that's simply not believable.
So, we need to begin by admitting that the moment someone knew how to rivet, weld, bolt, or otherwise bear weight — we made the prohibition against skyscrapers impossible. Every pyramid the Egyptians built, every leaning tower of pisa, every bridge over the river kwai, supports the obvious and incontrovertible development of skyscrapers.
Therefore, from a technological standpoint, it's impossible to the point of unbelievable that skyscrapers wouldn't exist in a modern world with the population densities you're talking about.
And before we continue, please bear in mind that as of the 2010 census, the United States had only four (4) cities that are as densely populated as you're suggesting: Guttenberg NJ, Union City NJ, West New York NJ, and Hoboken NJ, all of which are part of Metropolitan New York City. So what you're basically asking is how to have the population density of New York City without building up.
Why do we have skyscrapers in the first place?
Because they're cheap. But what drives that idea?
It's cheaper to build up than to build down. Fundamentally, once you hit bedrock, you're done for all practical intent and purposes with delving into the earth. Cracking bedrock is anything but easy, despite the benevolent use of high explosives. The removal of dirt means that dirt must be put somewhere, and while you could use it to expand your coastline (some cities like New York and Tokyo have actually done this) the reality is that it's expensive to move dirt around if you don't need to. On the other hand, given infinite technology, the sky (or, more accurately, the exosphere) is the limit.1
Humans, generally, don't really like the dark. We like sunlight, which means we like windows. Windows are cheap! Creating illumination conduits to bring honest-to-goodness sunlight into an underground room is really expensive! And if artificial illumination was all it was cracked up to be, we wouldn't make houses with as many windows both to save money and to save heating/cooling costs. etc. Yup, we like sunlight. Millions 'o years of evolution, there.
The shift from an agrarian society to an industrial society meant we needed a substantial increase in workers-per-square-foot of factory. Shipping those hound dogs in from a long way away is expensive, so skyscrapers were an obvious way to keep the workforce reasonably local. It didn't hurt that it also helped with the economy of scale for supporting industries like groceries and retail, and reduced the overall costs of policing, fire control, and other emergency services. In fact, it allowed for really big hospitals (which now can't use skyscraping techniques... that might be a problem) which meant less expensive and more available health care.
Finally, and kinda building off that last bullet, the beauty of skyscrapers is that it means you have access to everything! You have access to great shopping, entertainment, culture, education, because it's all nearby and therefore less expensive to access.
OK, so is it possible to build a modern culture without building tall buildings?
Assumption: I'm assuming that it's NOT OK to build down. In other words, a 100 meter building is a 100 meter building whether it's lifted to the sky or pushed into the ground. You don't clearly indicate whether the issue is the vertical space consumed by the building or simply the skyline you're trying to restrict. I say this because one way to meet your expectations is to dig down, but not carry away the dirt. Just spread it around, thereby raising the average ground level of the city, and thereby lowering the average skyline of the city. Maybe I even bring in extra dirt. I'm left with, say, a 500 meter building, but only 100 meters are showing above ground. It's still a "skyscraper" because the altitude above sea level of the roof is the same whether I covered the exterior walls with dirt or not.
And the answer is no. It simply can't be done. It's not enough for something to force builders to not build tall buildings because everything in the world would be pressuring them to build them: public demand, economy of scale, yada yada yada. Besides, your population density restriction absolutely demands tall buildings.
Look at it this way: the average thickness of the Earth's crust (land area, not ocean) is 30 Km. That sounds like a lot, but every foot closer to the center you dig, the hotter it gets and the more pressure you feel on the sides of the building that I've already assumed can't exist.3 But even if we can build "infinitely" down, you can only go so far. Think about the problems you'd have with earthquake abatement codes. And now, instead of slowly dropping sewer tank-to-tank,4 you're pumping it to the surface. You're pumping the CO2 up, too, because it's heavier than oxygen. And let's hope you don't need to deal with radon. Jeez, this is getting expensive.
So, unless we invoke TARDIS technology, there is no possible way for your civilization to achieve that population density without building up. Can't be done.
But, what if we weren't restricted by population density?
However, there is an option: cheap and fast transportation! Remember, the biggest reason those skyscrapers exist is economy-of-scale: you can get a lot of people where they need to be cheaply and easily. Business can access the workforce cheaply and the workforce can access business, shopping, entertainment, and education cheaply. If you made transportation cheap and easy, they could live anywhere and you could simply pass building ordinances restricting height, "for the beautification of our world and the enjoyment of the people. Allowing the setting sun to cast her joyous rays upon as many as wanting with minimal interference from artificial shadows."
1 Technically, given infinite technology, building beyond the atmosphere wouldn't be all that hard. However, the cost of a foundation for a building of that size, earthquake control that must compensate for the actual rotation of the Earth, the cost involved with pumping oxygen (and water) to those heights and carbon dioxide (and sewer2) down from those heights would probably sway the costs away from that height. So, for no better reason than I like the word "exosphere," I'm gonna stop building at the exosphere. (Sing it with me children! Exosphere! )
2 Yes, you'd need to pump sewer! Or, more accurately, you'd need to stop its flow periodically or end up with the proverbial sub-fusion amount of pressure at the bottom. We must deal with that today with modern mining, where water can't simply be put in a pipe and dropped a couple of thousand feet straight down. The pressure at the bottom would break any pipe and kill everyone nearby. It must be staged in tanks every couple of hundred feet or so to avoid that problem. Of course, you could have recycling every couple of floors, but now your costs just went up (no economy of scale). And just to make my point, I want you think really hard about the complications involving sewer solids at thousands of pounds per square inch of pressure....
3 You could change that with a clarification, but there are so many problems with digging down that it's really hard to believe it could replace building up. Wood rot, water seepage control, insects that evolved to dig through hard things, plant roots... It's easy to fix a broken building that's above ground... but fixing one that's below ground? Expensive! I wouldn't doubt that your story is basically built on the idea of subterranean buildings — but you might need to just "make it so" and stop worrying about how to justify it.
4 If that statement didn't make sense, you skipped footnote #2.