Suarez, who was found guilty of racially abusing Evra last year and slapped with a lengthy ban, apologised for his handshake snub in a statement on Sunday, having been branded a "disgrace" by United boss Sir Alex Ferguson.
Suarez's actions were the precursor to a feisty encounter between the two rivals, which United won 2-1. There was an incident involving both sets of players in the tunnel at half-time, while Patrice Evra was spoken to by referee Phil Dowd for celebrating in front of Suarez after the match finished.
The FA says the refusal to shake hands is not a disciplinary issue - and Dowd has reported that he dealt with Evra at the time, and that the tunnel incident was not serious enough to merit further action.
Statements from Liverpool managing director Ian Ayre and manager Kenny Dalglish revealed that Suarez had misled them by indicating prior to the match that he would shake Evra's hand.