After doing some research, by the later Moon missions the recovery fleet was down to two or three ships -- and officially only one for Apollo-Soyuz. I'm still pretty dubious that a carrier (really, an amphibious assault ship mostly used for Marines) was operating without the usual escorts, especially while conducting flight operations, even if they weren't officially "recovery vessels".

In terms of numbers, the Navy will be down to little more than 175 surface ships (i.e., not counting submarines), so if a dozen ships were involved with a recovery effort (whether officially designated or not) that'd be a significant fraction of the Navy, especially since that 175 includes ships not available due to repairs, refit, training, etc.

You probably do need something like an aircraft carrier as you are flying large helicopters off of it, and want to be able to conduct wide-ranging search-and-rescue missions if something goes wrong. You don't need a Nimitz-class carrier, but the large amphibious assault ships are what you want if you're landing out of helicopter range of land bases (which you would, to avoid a bad entry resulting in coming down on land).