The midfielder, 23, was playing away to Spartak Moscow in the opening game of the Russian Premier League season.
"When the match was stopped, the fans started shouting 'monkey' at me," said Ghana's Frimpong after the 2-2 draw.
"I just did not keep my cool, I showed my emotions and it was my mistake."
Frimpong was shown the red card after half an hour of the game in Moscow, with the score at 1-1.
He wrote on Twitter after the match: "A human being shouldn't be racially abused for the game that I love. I'm going to serve a sentence for being abused."
Ufa general director Shamil Gazizov said his club would not be calling for Spartak to be punished, describing it as an "unfortunate incident".
"There were people who could have shouted things. These are emotions which go away after the game. We are partners with the red and whites and are on good terms," he said.
"What Frimpong did was wrong. Sometimes you even have to hold back the tears and just put up with it."
Russia hosts the next World Cup in 2018, and Frimpong added: "Abused... and yet we going to hold a World Cup in this country where Africans will have to come play football."
Manchester City and Ivory Coast midfielder Yaya Toure, who has been racially abused while playing in Russia in the Champions League, has previously said  black players might boycott the World Cup unless the country tackles racism in football.
Last season, Dynamo Moscow's Congolese defender Christopher Samba was banned for two matches for gesturing to fans following racist abuse.
There have been more than 200 cases of discriminatory behaviour linked to Russian football over the past two seasons.
But in March former head of Russian football Vyacheslav Koloskov said too much is made of racism in the country and has also questioned why monkey chants are assumed to be racist.
Former Arsenal player Emmanuel Frimpong was sent off playing for Russian side FC Ufa after he gestured to the crowd with his middle finger in response to alleged monkey chants.