I have taken the heads from deer I have shot and did some tests on them. From 50, 100, and 200 yards shot with a single shot on the head. Every shot from head in and 90 degrees was a complete pass through with massive damage. I have zero doubt a well placed shot in the head would bring down a deer, though probably not with a Gamo. And yes I did my tests with a 40 grain 22lr, but with slugs anymore you can get the same energy dump.
Have talked to many old hunters, like 80+, and they have all stated that when they were younger taking deer with a .22lr. Basically that was all they had and needed to take the deer to feed the family.
Though you need to know your state laws and if illegal don't shoot live deer. I don't think it it legal in any if the states to take deer with a .22lr or airgun.
A lot of people need to understand that many of the hunting laws are do to poor hunting practices. There is no need for a 7mm or the like to hunt deer. The larger calibers help crappy shots anchor animals with poor shot placement. Bigger caliber means more wiggle room with poor shot placement.