He will initially have no rights at all as he would have to take you to court and file for paternity (assuming that you don't allow him to be at the birth, don't allow him to sign the birth certificate and don't allow him to sign the paternity form that legally establishes paternity - refusing these things is all within your rights). Once he files for paternity, that will open up a whole can of worms regarding child support, custody and visitation. All of this takes time - he will have to file a motion or complaint to establish paternity, you will have to be served, the court will need a return of service to prove you were served, then they will set a date to review the motion or hear the complaint. All of this takes months. Once you get served, you would call your state's child support enforcement agency and they would take over, sending you paperwork, him paperwork, receiving the paperwork, assigning one of their attorneys, notifying him or his attorney (if he has one), scheduling a date, etc. I filed for child support 10 days after my son was born and it took 8 months to get to court, so the slow wheels of justice will work in your favor. By the time you get to court to establish paternity, decide on a visitation schedule, child support, health insurance etc. your baby will be older. At that point, a judge may very well decide that the father can have overnight visits or visits longer than a couple of hours and if you have to, you can pump and your baby will be fine (moms who use childcare pump every day and still breastfeed when they are with their babies).
Anyway...read up on the laws in your state - all of this info is available on-line. You don't have to worry about anything right now. Let him talk all he wants...nothing he is saying will apply right after the baby is born so don't let him bully you or stress you out. You focus on your and your baby and deal with his requests in court if/when you have to.
@ Cheryl B you can't go to court to do paternity, support and visitation for an unborn child. There is nothing that she or he can do until the baby is born. The only time pre-birth orders go into effect is in states that recognize surrogacy and in the case of adoption, and in many states even in those instances, not a piece of paperwork can be filed until the child is actually born.