If a jeweler damages my ring, can they keep it while determining what compensation I will receive? Can I force them to give it back and have them pay for repairs?

There are two different issues here:

The first is physical control over or access to the ring. The answer to that is, it is your ring, that is, your property. The store, the jeweler, or indeed, anybody else, has no right to tell you what to do with it or keep it against your will. If they refuse to return your property on demand, you might consider contacting the police. Keeping someone’s property without their permission is theft.

Second is how to compensate you or remedy the situation. Without going to court (see below), neither you nor the jeweler can unilaterally determine what is fair or appropriate compensation. Both of you, of course, are entitled to your positions or opinions; however, until and unless there is a court order, neither of you can force the other to accept your position. You can and should try to work matters out between you, since court can be expensive, whether suing or defending; however, at the end of the day, if you cannot work out a reasonable settlement of the issue, you have two options:

1) If you have a homeowners or other policy which insurers the ring, you may be able to submit a claim for its damage to your insurer. If the claim is accepted, the insurer will compensate you and then decide whether it wants to recover against the jeweler. You’d be out your deductible (which you could still sue for later, such as in small claims court), but would get the bulk of your compensation. If you have insurance, check your policy, not just for coverage but to see what your obligations are to report the loss to the insurer. (Often, you need to report it fairly quickly.)

2) If you don’t have insurance covering the ring, don’t want to submit the claim to insurance, or your claim is denied (e.g. you violated any provisions regarding how and when you need to inform the insurer), then you can sue the jeweler for compensation. In a lawsuit, you would be primarily seeking economic recovery, such as the amount of money necessary to repair the ring.