Me of the past would have, if they had an online presence. Not so much out of an interest in him as because she was kind of into cyberstalking back then.
I don't have any kind of contact with the ex, except that he lives in the apartment complex right next to the one where my dad lives, so if dad runs into him, he's liable to tell me about it. I admit I felt a bit of schadenfreude when I heard he was married with a kid and a second on the way. Sometimes - very rarely - I wonder if he thinks of me as "the one who got away."
Keeping in touch with an ex you parted with on amicable terms could be healthy. My breakup wasn't that kind.
The relationship was already dead; he just waited for me to be the one to break up, claiming it was to protect me from the pain of being dumped, but I suspect he simply did not care. Living together was simply a convenience to the both of us. I had someplace to live, he didn't have to do all the cooking and cleaning by himself. Near the end, we were more like roommates, and I felt a little better after I moved into the other room to sleep on the couch every night. When he was gone on vacation and I realized after a couple of days that I felt relieved that he wasn't around, I finally admitted to myself that there could be no other option than to break up.
There can be no value in me cyber-stalking him, other than to laugh at his misfortune, and that I do not need. I am better off now, and whether he is better or worse off.. simply does not interest me very much.
Now, if I had an ex I still liked, the issue might be different...