Hi Katagat, and welcome to the boards.
This seems pretty unlikely, for a couple of reasons.
First, most infections you're probably worried about aren't transmitted through urine. Most STIs are associated with the genitals, and won't be transmitted through a small amount of urine that ends up on the seat.
(side note: It's actually a common myth that urine is sterile. Urine is full of bacteria - but not ones that cause issues, for the most part. Everyone has a microbiome in their bladders, just like we do in our guts, mouths, on our skin...etc. So bacteria are normal there. You can search "bladder microbiome" if you want to learn more!)
Second, you can't get an infection of any kind unless you get exposed to *enough* of the pathogen. A bit of urine on the seat, even if it did come from a person with some kind of infection, is very unlikely to have enough of a virus or bacteria to then transmit the infection to you. That's especially true because you soaked it up with toilet paper first, so much of whatever was in the urine will have gotten trapped in the toilet paper.
In sum, you don't need to worry. If you still do feel concerned, you can always get screened for STIs after a few weeks have gone by.