I have a house built in the 1960’s, and when I pulled off the plaster as part of the kitchen remodel, I discovered 2x2 studs against the gray cinder blocks. My question is, should I cut out the 2x2s and reframe with 2x4s, or can I furr out the 2x2s another 2 (1.5) inches,so I don’t have to reframe and can meet code?
Those 2x2s are fine as furring strips over block. They are really more a part of the wall finish. The block is the structural wall, it can stand alone without the 2x2s and drywall.
If you are going to mount cabinets on these furring strips, I think I'd top the furring strips with 3/4" plywood.
If you want a deeper wall for insulation, plumbing, etc., @manassehkatz mentions it would be smarter to sister 2x4's side by side with the 2x2's than to extend the 2x2s or try to attach 2x4's on edge. However it may be even easier and stronger to just remove the 2x2's and build a full 2x4 wall with top and sole plates rather than just furring strips.
As far as I know, and as already noted by @batsplatsterson, the cinder blocks are the structural portion of your wall, so for purposes of providing proper structural support, the 2x2s are not necessary at all.
But there are other factors:
When I redid my kitchen ~ 20 years ago, part already had 2x4s over brick/block and part did not (used to be a doorway) and I built out with 2x4s and added insulation. If the walls you are working with are exterior walls then one advantage of a deeper wall cavity (i.e., 2x4 instead of 2x2) is the ability to add more insulation.
Electrical and Plumbing
If you want to install any electrical (wires/receptacles/switches/lights) or plumbing within the walls as part of the renovation, deeper walls will make that a LOT easier. For example, with 2x2s + 1/2" drywall, you are limited to a 1-1/2" or possibly 2" deep junction box, but with 2x4s you can put in deeper boxes without a problem.
2x2s should be strong enough to hold wall cabinets, but I'd feel more comfortable with cabinet screws going into 2x4s. You can use longer screws - a 2x2 is actually 1.5"x1.5" so 2" screw (including the 1/2" drywall) is your absolute max. with a 2x2, plus 2x4s are going to be stronger no matter what size screws you use.
So I don't think this is a code issue per se (but I am not a code expert at all), but if you can spare the extra 2" all around then I think moving up to 2x4s is worth the effort.