Oh, the pain...
Bedrock is your first problem. You're not building an itsy-bitsy building like the Burj Khalifa or the Tower of Pisa, your'e building the building, the biggest, honkingest, Oorah-est building on the planet. And you're guaranteed to crack the foundation if we don't go all the way down to bedrock, grind the bedrock flat, drill in a bazillion holes for metal rods, and lay the thickest foundation the world has ever seen ... because you want to build up, not out (in a area that's just begging for more urban sprawl), which is probably a horrible idea, but let's roll with it.
So, let's examine a geological map of Yuma County, AZ. We'd need someone like Arkenstein XII to give us the low-down, but I'm betting the word "Granite" is good (My thanks to Ash for pointing this out), which is a good thing, because there's more of it than basalt.
Unfortunately, what you'll quickly discover looking at that map is that while you'll have good sun coverage for your solar panels, it's pretty much the worst possible place on the planet for a foundation the size of what you're looking for. The situation isn't as bad as I feared, but it's still inconvenient. But, there's spots out there we can make this work because we're about to calculate our footprint size.
If I remember my geology classes in high school correctly, the areas of the world that would be best for a honking huge foundation are pretty much the worst for solar panels. I could be wrong, though. High school was sometime during the Cretaceous Period. But it's worth thinking about.
Water is your second problem. You want to be self-sufficient for a billion people. This is actually a really ugly calculation. People need to drink, dispose of waste, experience hygiene (in a sealed system w/1B people... oh, yeah), then there's crops, industry, humidity control, yada-yada-yada. I'm going to take a complete pull-it-out-of-the-air guess and suggest calculated per-person you need something like 100 gallons of water per day. Let's pretend we have 100% efficient recycling and that the recycling is capable of processing every drop of water each day (an oversimplification that literally will make angels weep). That's 100 billion gallons of water or a storage tank 13.4 billion cubic feet in size. If it's just half the height of the aforementioned Burj Khalifa (1,358 feet) it covers a little over a third of a square mile. Let's call it a third for convenience and just build a little higher.
So, a third of a square mile, doesn't sound too bad, and we should be able to find a nice, big chunk of basalt even near Yuma we can work with. Let's pour that sucker a solid 30 feet deep, fully reinforced and tensioned, of course, and another 2,000 feet (easy) of honeycombed water tank so we can build a support structure on top of it. It's the honeycombs needed to support the rest of the building that causes the height (which is a round guess) because they consume volume not previously accounted for.
And it's worth noting that if you didn't have a crisis driving this process before, the diversion of water to fill that tank will cause it. OK, maybe not really, but the resulting fight between California and Arizona over water rights would likely start a second civil war in the U.S.
Food is your next problem. This fascinating article boils the issue down to a simple number: one acre per person if you do it the "natural" way (growing crops, grazing cows, etc.). That's (*cough*) one billion acres under your roof. To give you a sense of size, The United States is 2.3 billion acres. To make things interesting,
just over 1 billion of those acres are devoted to crops and grazing. Considering the potential for U.S. food production, that casts a shadow on that previous article's conclusion that 1-acre-per-person was needed, but we'll roll with it. So, you need to build a building (now it has to be vertical) that houses the entire food-production land usage of the United States. That third-of-a-square-mile is looking a bit small, as is my 100-gallon-per-person estimate.
And something must be wrong with my sources at this point. But not too wrong. People have been complaining about too-many-people vs. too-little-arable-land for a long time now. This might simply be drawing a huge underline below the problem. In other words, unless you can compress food production something awful, your premise isn't believable as you need a planet to feed a planet worth of people.
Let's assume that you've improved food production technologically such that, somehow, you need only 1/1,000th the space to feed a person. Thus, one billion becomes one million ... acres (sigh). A third of a square mile is about 211 acres, so you need 4,740 floors for food production at about ten feet per floor (which is incredibly unrealistic) so now your building is working on 50,000 feet of altitude (which is well above the 35,000 foot average cruising altitude of a Boeing 737) and we haven't even begun to address industry and infrastructure — much less housing.
I did say building up, not out, was a horrible idea....
And it's time to stop
Yes, you can lower the height of the building by spreading out the foundation, but that causes problems with a believable foundation (remember that geological map of Yuma County). Obviously we can't conceive of a believable way to do this today. So I'm going to go out on a limb and suggest...
Your building's foundation is the size of Connecticut and the building is approximately 20,000 feet (6,100 meters) tall.
And we won't talk about what the weather around this sucker would be like.