Yes, it's perfectly fine to leave a bottle of red wine in the fridge if you can't finish it that day. In fact that is the best way to preserve an open bottle if you have no other way. You can extend the life of an opened bottle of red wine by a couple of days by sticking it in the fridge. Room temperature red wine starts to get funky around day 3 while in a fridge it might be 5 or 6 days. It really depends on the wine.
The only problem, unless you like cold red wine (if you've ever been to Paris, you know that's how they serve it), you will need to let it sit out in a glass until it warms up. Make sure you put a cork or some other stopper back in the top of the bottle to slow down the rate of oxidation. Personally, I like my red wine served at cellar temperature (around 50f). I think it makes the red wine easier to drink. Nobody likes hot red wine unless you are making Gluhwein.