Yes, most air rifles of the same make and model use the same power system between calibers with a few exceptions in the higher priced and larger calibers.
However, I question the practicality, a new barrel/breech block, and sights, if one is available is likely to cost close to what you can buy a new Crosman Vantage .22 air rifle, the going price at most air gun venders is about $119.00, and what's better than a Crosman Vantage, two Crosman Vantages in .177 and .22 calibers.
For what it's worth I have a Vantage in .177 caliber, when I purchased it I wanted it in .22 cal. but it wasn't available, so I took it in .177, after smoothing the trigger and shooting it I was happy with it the way it is, I have other air rifles, some in .22 cal., but I wouldn't trade my vantage for anything else.
Good luck with whatever you decide.