It doesn't sound much like it. Do you have any indication if the baby was considered early, on time or late? If you got her pregnant from the sex on September 8, a full-term baby (40 weeks GA) would have had the due date of June 1. This makes it sound like her baby born on June 15 would have to be two weeks late to be yours, but that isn't quite true -- sperm can live in a woman's body 5-6 days, so it would only actually only have to be 1 week and 1 day overdue to be in range to be from the sex with you. But even so, it is quite unusual for doctors to let a pregnancy go even a week overdue (especially if it was a first pregnancy).
Go ahead and get a DNA test if you want her to stop talking about this to other people. Get the name of a reputable lab by calling the family courts in your area of jurisdiction and asking the judge's clerk for a list of labs accepted by the court for determining paternity. Be sure to go in person to the lab with your ID, and it's not a bad idea to go with her and the baby so you can both witness who gets swabbed. In short, use a real lab that has protocols to prevent fraud, not one of those drugstore tests.
One of the prices men pay for living in a world of free sex practically any time they feel like making an effort to get it, is that sometimes they have to take a DNA test. Handle it as such, and remember that in your grandpa's day he would think you're livin' the dream. And remember, condoms are your friend. A DNA test is not the only thing you might get someday.
At this point I think the only hint that will truly put your mind at ease is a DNA and they are very cheap now that the child is born