Shoddy work from a roofing company. What recourse do we have?

Ask them politely to come back out with YOUand look at. Maybe they'll fix or give you a reasonable explanation as to why they won't. You can report to BBB at the least if you still aren't satisfied or take to small claims court....

I would just hire a skilled carpenter or fix it person and have them make the needed repair before you get a leak and bigger problem. Ask for references. You can go to lowes or Home Depot if you don't know where to start and they can provide a list of reputable contractors in your area, still get references on work they have completed.

Did you mean to say "reinstall" the gutters correctly? So you are saying the roofing company removed and then reinstalled the gutters? If they did that and they reinstalled them incorrectly, then it is on them.

If there is a problem with the gutters themselves, not the installation, then it goes to how long the gutters were guaranteed for.

If it is in the Northern part of the US and it is just the gutter pulling away from the roof, it's because of the winter we had. Mine are pulled away in two places. It is from Ice. Need to call someone with a ladder to hammer the spikes back in.
Either the roofing company or the gutter company, but they will want to be paid to come out.