Rear brake pads rubbing rim?

One of the caliper mounting bolts is either missing or it missed the threaded hole. Once installed correctly, the end of the anti-rattle spring, (blue arrow), must go under the mounting bracket, (red arrow).

More likely you removed a cast aluminum wheel and installed a spare that's a different steel wheel. Unless there's something I'm not seeing, this didn't just start with the spare tire. That caliper can't lift up unless the one mounting bolt is missing or mispositioned. Try to pry that end up more with a flat-blade screwdriver. If you see any movement, it proves that bolt isn't doing its job.

I'm actually dealing with this right now with my Ram truck. My friend owns a body / repair shop. He repaired damage from a deer hit, and while he was at it, he replaced the brakes on all four wheels. After many years of preaching, I can't convince him to use a torque wrench. Ends up, by the time another friend delivered my truck, (30-mile drive), one of the front caliper bolts had fallen out and the caliper was rubbing on the wheel. I got it back down, then properly tightened the one remining bolt. That is holding it in place for now, but I need to visit a salvage yard to find a replacement bolt. The alternative is to walk the 30-mile route and look for the old one!