Without naming Townsend, Latvian Ostapenko wrote on Instagram: “Hi all – I wanted to apologise for some of the things I said during my second-round singles match.
“English is not my native language, so when I said education I was speaking only about what I believe as tennis etiquette. But I understand how the words used could have offended many people beyond the tennis court.
“I appreciate the support as I continue to learn and grow as a person and a tennis player. Goodbye New York and I look forward to being back next year.”
The incident occurred at the end of the match when the two players met to shake hands.
Ostapenko, 28, confronted Townsend for not apologising for a net cord, which had occurred during the first set, then accused the American of having “no education”.
A net-cord winner is when a player wins a point because their shot hits the tape of the net and lands on the side of their opponent.
It is not a rule to apologise for a net cord, but some players feel it courteous to do so.
After the confrontation, on-court microphones picked up 29-year-old Townsend telling Ostapenko to “take the L [loss]”.
Townsend, the world number one doubles player, then whipped the crowd into a frenzy by celebrating with them wildly.
Boos rang around Court 11 as Ostapenko departed.
Ostapenko, who described Townsend as “disrespectful” on Instagram after the match, posted later on Wednesday to deny accusations of racism.
“I was never racist in my life,” she wrote.
“I respect all nations of people in the world, for me it doesn’t matter where you come from.”
Ostapenko’s US Open came to an end on Thursday, when she lost a doubles match with Barbora Krejcikova.
She failed to fulfil media duties afterwards, citing medical issues.