Probability is 100% you're the father, though it doesn't have anything to do with sex on May 14. It has to do with the "and so on" as you called it, after May 14. If you are the only guy she had sex with since her mistake on May 11/12, and if you continued to have sex with her through May and into June, hey, you're the dad.
Here is why, and this will be a little long, but please attend. Basically, the doctors said something to you in medical terms that don't mean what they sound like in plain English.
The count of a pregnancy used by doctors and nurses is called GA, which stands for "gestational age." It is the count in weeks, like 9 weeks "pregnant" or 9 weeks "along." This might sound to you and me like they are saying "9 weeks since conception" or "9 weeks since the sex that caused the pregnancy." But that is not what they are saying. When a medical person uses a number for pregnancy in weeks, it means "since the first day of the woman's last period." That starting point is around two weeks earlier than conception.
Medical science uses the first day of the last period as the start point of the count because it's a big, obvious signal, and in the days before ultrasounds, it was the only signal they had. (A woman would see her doctor when pregnant and ask how far along she was. The doctor would ask when her last period came before she got pregnant.) They began to calibrate the count of weeks to begin with day 1 of the last period, even though they knew a woman having a period is not yet pregnant. It was the only thing (before blood tests and ultrasounds) that they had to count from.
That kind of count of the length of pregnancy is still used today by all medical people, ultrasound techs, textbooks, Lamaze teachers, "What to Expect When You're Expecting," and the whole medical world (despite more modern ways being available to assess when conception actually happened). The GA time frame starts on day 1 of the last period the woman had before she became pregnant, and goes 40 weeks. A full-term pregnancy from conception to birth is only 38 weeks long. The first two weeks of the 40-week GA count are to back things up to the last period so the count can begin with it. Conception doesn't happen until around day 14 of the GA count.
For you, this means that if your girlfriend's doctor, nurse or ultrasound tech told her she is at almost 9 weeks, conception was about 7 weeks ago. As of today, July 22, 7 weeks ago was June 3. This fits with you two being told earlier that conception was around May 30 or June 1, since you did say "almost" 9 weeks and not "exactly" 9 weeks.
If conception was around the first of June, sex earlier in May didn't produce the pregnancy, not even May 14 with you. It would have been sex (presumably with you and no other candidate) at the end of the month.
You can check what I'm saying by going over it with the doctor. Just be sure not to fall back on phrases like "How far along is she?" because all you will get is more stuff using the GA, and that confuses things. Either ask when conception was, or if that's too direct, ask for the baby's estimated due date. If the estimated due date was based on ultrasound evidence, you can use it and count back from it 266 days on a calendar, and that will give the estimated conception date.
She did NOT get pregnant from the earlier May intercourse. She got pregnant around May 30th. Sperm cannot live from the 12th to the 30th. You are the baby’s father.
This is so confusing. Brooklynguy, I don't see where you say you got a DNA test. Has that been edited out? Did you go with her to the sonogram?
I don't know what's going on. Is that standard - due to COVID? Or did she tell them quietly she didn't want you there?