No, he doesn't have that "right." He is intruding into YOUR life at this point. He has a right to know that she is loved and safe (she is), if she has a medical or educational issue at any time, if she is going to be out of state for any reason... I can even see him getting on the phone (speaker phone if needed) to talk to her once a day, maybe say goodnight, even read her a short story - THAT is a connection that may be meaningful to her at this point. The other stuff is just controlling you. You don't work for him; you are not the daycare provider (and even a paid provider would get pretty pissed if they got 5 phone calls during their watch!)
Divorce is hard in parents - but that is life. I can't imagine calling my x up to 5 times a day when the kids are with him to ask what they are doing or to "kiss them for me." Yes, mine are older - but it is HIS time with them. I don't want him infringing on his time with me, either.
Now, I understand that your x screwed up and has supervised visits - guess what, HIS consequences for his choices.
Maybe you can have your cell plan changed (or just suggest that it was changed): "Sorry ex, I can't accept text messages anymore - you'll have to use email or call with emergencies only, besides this one time a day that I can make sure that baby is available for you to talk to on the phone."
Ugh, I know that you still have deep feelings for him, but he is just a messed up controlling creep from my vantage point. And you deserve SO much better!!!!
