If rusting is your main concern, could you paint it with a rattle can? Black might be easier to get a decent color match, but then again it is on the underside anyway.
To answer your question and seeing your location, yes, it will rust. It will be surface rust IF the car is driven daily and is exposed to moisture or road salt. It will take years for that surface rust to be of any real concern (such as rot through).
Look online look for The product called ‘POR15‘. You can get it in spray of brush on. Clean the surface, apply the POR15 and everything will be perfectly fine.
What did you do, OP? Ran over a concrete parking stop? Ouch! Depending how hard it hit underneath, you might want to check the exhaust upstream for leaks/cracks. For the other stuff, you can just apply an undercoating and be done with it. I have one that is rubber-based, rather than asphalt, which smells like crap.
In addition, you might want to check the wheels and tires that hit, and if it was anything over a few MPH, I'd check the alignment too. Good luck.
Try POR15 or any type of rust preventative paint. Just make sure you prep properly and it will last a long time. POR15 sells kits with prep and paint stuff.