If you have any concerns, don't belt up, see a doctor.
Next step: Find out why you hurt your back doing leg press. Namely, what did you do wrong with your positioning? What mobility and stability issues caused you to have improper positioning? Once you find out those things, you can take steps to ensure it never happens again. Otherwise it will happen every week like clockwork. Trust me.
I would NOT belt up for the sake of moving weight. When I threw out my back squatting, it was so bad I couldn't walk for a few days, I started over with just the bar and I packed my belt away so I wouldn't be tempted to lean on the belt for the extra poundage. It took me almost a month to be comfortable deadlifting anything over 135, but two months after that, I hit a 500# pull again.