No you cannot. Sorry but the dimensions of the the S&W 15-22 are slightly different just so you can't do that. As the 15-22 is a completely polymer gun, I think it doesn't have the strength to handle .223/5.56 ammo. Awesome rifle though, I love mine and hope you'll enjoy yours. If you need an inexpensive way to get an AR try building one. I've been checking prices and I think you can do it if your very careful how much you spend on parts for about $650-$750. Maybe less if you get some great deals. There are a lot of good DIY videos on building AR's on youtube if you want to see how to do it.
Guns
Absolutely, watch classified forums for parts and you can probably go even cheaper. An example, whenever I build a new rifle for myself I order a Daniel Defense lower parts kit but I never use the safety selector in it, I always put in an ambi. It's still cheaper to buy the kit so I usually list the selector really cheap in the classifieds and have also put them (and seen others do it too) on the pay it forward thread where you just pay shipping at the most. A lot of people do the same with triggers too. I just sent off a bunch of free spare parts to someone who was wanting to do a really reasonable first build, basically all he had to buy for his lower was a bolt catch and some springs and detents, maybe $10 worth of stuff from J&T or model 1 who are probably the best for individual small parts.
I don't really have any loose small lower parts left right now but probably will again by the end of the month. If you don't get started right a way send me an IM and I'll see what I can put together. Check gunbroker, there was a company that was selling M&P lowers that were cosmetic seconds, perfect tolerances in the machining but they had a little flaw in the laser engraved logo, for $79. Depending on where you live if you have gunshows etc you should be able to find a lower for no more than $100. It's very easy to find standard - new take-off M4 stocks for sale dirt cheap. You can get a standard A2 pistol grip free (I've got a few left if you want one). You should be able to put together a complete basic lower for under $200 maybe close to $150 if you work at it. If you check AR15 classified ads you should be able to find a lightly used complete upper in the $350 range with everything.
Short of that DSA has really good prices on their basic uppers and they're better quality than a lot of others on the market. $400 should get you a new complete upper. You can build the upper yourself and save maybe $50 but unless you have some experience I wouldn't suggest it on your first build. Unless you understand headspacing (and have a gauge) you'll have to take it to a gunsmith to make sure it's safe. unless you're going to build more than a couple rifles the headspace gauge or the trip to the gunsmith will eat up your savings. Some people will tell you you can headspace with a dummy cartridge or snap cap but I don't feel safe doing it that way, to each his own.
The lower construction is pretty idiot proof. If the hammer drops when you pull the trigger, the safety lever functions and the mag locks into the magwell you're most likely safe. there are a lot more things that can go wrong with the upper so there's no shame in getting one that was professionally assembled for a few dollars more. Good luck.