Referee Howard Webb did not see the incident during the game, but on reviewing television evidence he did not deem it worthy of a sending-off offence, so no further action will be taken.
No action will be taken either in regard to Newcastle midfielder Cheick Tiote, who appeared to stamp on Tom Cleverley's leg, or to Stoke defender Robert Huth, who was involved in a similar incident with Liverpool forward Luis Suarez.
In both cases, the incidents were seen by officials at the time and therefore no further action can be taken.
Newcastle manager Alan Pardew said after the game that he felt the Van Persie incident warranted further investigation, saying: "He has looked at Yohan and he has elbowed him, and I think that perhaps needs to be looked at, if I am honest."
He added: "There's a bit of history from last year and I don't know if Robin has been caught up in that, but that was just a little bit unsavoury.
"I actually thought Yohan wouldn't have gone down if he hadn't done something.
"I didn't see it myself, but I have obviously seen it on the replay and it looks like he has looked at him and put his elbow there.''
The "history" referred to by Pardew was an incident involving Van Persie, then an Arsenal player, and Newcastle's Dutch goalkeeper Tim Krul which took place last season.