Sit ups should not cause redness of the skin of the navel, unless you were wearing tight fitting clothing that was rubbing against the navel. They may cause some soreness of the abdominal muscles, but this is usually more generalized and not just localized to right around the navel.
I think it is more likely that you have a minor infection of the skin of the navel. If the area is not very warm or painful, and if the redness is not spreading, then you may be able to control this with good skin hygiene and anti bacterial ointment. If you have a navel piercing, you should definitely remove that to help things clear up.
However, if the redness is spreading out onto the skin around the navel, if the area is very warm or painful, or if you have any fever
, or if there is any discharge, these could be signs of a more serious skin infection and you should seek immediate medical care. A serious skin infection is called cellulitis, and cellulitis does often require prescription antibiotics in order to get the infection under control.
Please see your doctor
right away if this description fits your case!