Van Dijk's dismissal at Newcastle Utd was his second in 231 Premier League outings
The defender was sent off in the 28th minute for fouling Alexander Isak.
Van Dijk was banned against Aston Villa on 3 September and will be unavailable to face Wolves on 16 September.
He appeared to remonstrate with referee John Brooks prior to walking off the pitch at St James' Park.
Netherlands centre-back Van Dijk, 32, also appeared to argue with the fourth official before making his way down the tunnel.
"The defender admitted that he acted in an improper manner and used abusive and insulting words towards a match official after being sent off, and the sanctions were subsequently imposed by an independent regulatory commission," said an FA statement.
Writing on Instagram Van Dijk said: "I'd just like to make it clear I totally accept the panel's decision.
"It was my first red card of my Liverpool career and I let my frustrations get the better of me in an intense and heated moment.
"I apologised immediately after the game to the match officials and take full responsibility."
Referees' chief Howard Webb has said he is "determined" officials would not "turn a blind eye" to bad behaviour from players and coaches this season.
Webb, speaking on Match Officials: Mic'd Up, backed the decision of Brooks to dismiss Van Dijk.
"We think it's a good identification of a foul by John Brooks the referee," said Webb.
"We see Virgil van Dijk does play the ball, but to get there he clearly kicks through the foot of Alexander Isak.
"It's not only a free-kick but it also denies Isak an obvious goalscoring opportunity. He has to be sent off in this circumstance and the VAR checks it."
The incident occurred three minutes after Liverpool went behind to Anthony Gordon's opening goal and with the visitors under pressure in the game.
Uruguay striker Darwin Nunez came off the bench to score twice late on and hand Liverpool a 2-1 victory.
-- Courtesy of BBC Sport