No. The chances of getting an STI from peeing in a public bathroom are in the incredibly low-to-no range. Could the organisms that cause chlamydia and gonorrhea infections be present in people's urine that's splattered on the seat? Yes. But are those organisms going to find their way into your urethra or vagina? Not unless you rub the tip of your urethra (or vaginal entrance) on a contaminated toilet seat. Who does that?
Having worked in a family planning clinic, every STI case I ever came across was transmitted from sex. STIs are passed primarily through oral, vaginal, and/or anal sex. Unprotected anal sex with an infected partner poses the highest risk of infection, then unprotected vaginal sex, and finally unprotected oral sex. And while condoms offer strong protection against HIV, chlamydia, and gonorrhea, they are less effective at protecting people from passing syphilis, herpes, and the human papillomavirus because condoms can't possible cover all of the possible sites of infection, such as a person's pubic mound, a man's scrotum, or a woman's labia. As such, regular STI testing is recommended – especially if you're in the market for a new partner or think your partner has other partners.
Dr. Debby Herbenick, author of Sex Made Easy and Because It Feels Good: A Woman's Guide to Sexual Pleasure and Satisfaction, is the Co-Director of the Center for Sexual Health Promotion in the School of Public Health-Bloomington at Indiana University (IU) where she is a Research Scientist. She is also a sexual health educator at The Kinsey Institute for Research in Sex, Gender and Reproduction where she writes (and hosts audio podcasts of) the Kinsey Confidential column and coordinates educational programming. She has a PhD in Health Behavior from IU, a Master's degree in Public Health Education (also from IU) and a bachelor's degree in psychology from the University of Maryland, College Park. In addition, she is certified as a Sexuality Educator from the American Association of Sex Educators, Counselors and Therapists.