There are two sides to the "snoop". I agree with Hangn, if there is reason to suspect and you are in a serious relationship then - yes. Think your partner is sleeping with someone else? I'd say that's something worth finding out.
In this situation its inappropriate for a myriad of reasons. What has come out of snooping is "sympathy" for the ex. If he doesn't pay support, then file your contempt and move forward.
I think its normal to be curious about the father of your child. I would advise to throw the phone away - you'll be happier for it and won't get anchored with sympathy OR fear worried about what's happening.