Full frame, or FX, light sensitivity is MUCH better than any DX currently, so it depends on the lighting conditions where you shoot. I do sporting events with the D7100 here and there, and it's a champ. BUT (everywhere I go...) I only shoot in daylight and/or VERY well lit conditions. A full frame should get you more than a full stop of low noise ISO performance so you can crank it a bit higher and keep the shutter speed that you need, even when light isn't quite as favorable.
Then, comparing the D610 to the D750, I'd say go D750, just for the extra cross-type focus points (faster AF when you have those moments that can't be lined up perfectly). Combine that with some nice glass, and you should be set. Having said all that... most ice rinks I know of, I wouldn't be afraid to shoot the D7100 at all, and most rugby matches that *I* ever get to come across are held in full daylight, but there aren't any "stadium" matches here, so not sure what the lighting would be there.
Any place you could rent a D7100 and/or one of the FX bodies for a day to compare? That would probably give you the best answer for how it performs per your expectations.