When we say "skin to skin" contact for STDs, we mean direct mouth to genital, genital to genital, or genital to anal. The only STDs that are spread by skin to skin contact are herpes, syphilis and HPV. The skin around your belly button is too thick for any of these to penetrate and infect.
Perhaps there was something in the hand sanitizer that you were allergic to, or irritated your skin, or even in the laundry detergent he uses that got on your skin when he used his shirt to clean you up.
Is the skin just red or is it broken and irritated? If it's just red, I'd leave it alone for a bit. Don't use anything on it other than just cleaning it normally. If it's broken skin or has any blisters, you may need to see your doctor.
Also - just a note - using hand sanitizer after any sexual contact isn't going to help prevent an infection. Hand sanitizer has alcohol and often perfumes and things that can be very irritating to skin. Showering isn't going to help, either. It makes you feel clean, and that's great, but it won't prevent infection.