Can I use pressure-treated 2x2 posts as a base for an aluminum arbor?
Wood that thin doesn't offer much in the way of strength or stiffness. Even if you built your arbor as a completely self-supporting structure and only needed the posts to anchor it, 2x2s would offer relatively little in the way of storm resistance, for example. 

A better approach is to design your arbor so that it's self-supporting and rigid, and mount stainless steel brackets to the slab using suitable anchors. Your arbor's legs would attach to the brackets. They won't offer torsional stiffness, but they'll keep the arbor in place. 

You could also embed pipes in the slab, into which you'd insert your aluminum legs. These could be PVC. You'd then use shims to secure the legs of your arbor into them. The pipes should be open at the bottom with a drainage basin of sand or washed rock.