Answer: If the stem or vine is crushed then it will not be able to send water and nutrients from the roots to the part of the vine out beyond the damaged spot. You will know quickly because the foliage beyond the break will wilt.
If you have ripe fruit on the vine out beyond the damage, you can pick it by cutting it off the vine. If it is not ripe however it will not ripen after being picked. Melons that are ready sometimes "slip" off the vine, otherwise you can check the bottom of the melon. On a ripe melon, the spot where it touches the ground should be yellow instead of greenish. A ripe melon can be kept for a few days or up to a week at cool room temperature and this can sometimes make it taste better. (Refrigerating it will cause cold damage.)
If the melons are closer to the roots than the broken spot, then they should be fine where they are and can be left in place to continue to grow and ripen. You can trim off the dead vine. I hope this helps!
Q&A Library Searching Tips