Tarpischev labelled Serena and Venus the "Williams brothers" on Russian television last week, leading to a fine and one-year ban from the Women's Tennis Association (WTA).
WTA chief executive Stacey Allaster demanded an apology and Tarpischev delivered with a statement on the RTF website.
"Serena and Venus are without doubt top sportswomen and they have played tennis to the highest possible level, perhaps even a different level," Tarpischev said.
"The Williams sisters will always stand out on the tour, as they do in the sport thanks to their talent and successes. They have managed to set new standards and results that will become the new benchmark for other players.
"My comments were spontaneous and happened after we went live on the 'Evening with Urgant' show and they not only put tennis in a bad light but also drives a wedge into the strong and friendly tennis family within our sport.
"I did not want to cause any offence or make fun of any of the sportswomen. I am sorry that the joke I made, when translated into English was taken out of the context of a chat show and so much attention has been brought towards it.
"I don't think that the incident needed to be blown out of proportion. Everything that was said on the TV show was said without meaning to cause offence."
World number one Serena has won 18 grand slam singles titles and 13 doubles crowns – with the latter all won with sister Venus.
Venus is a seven-time grand slam singles champion.