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.
“There’s no beef,” added Townsend.
“But I didn’t back down because you’re not going to insult me, especially after I carried myself a certain type of way, with nothing but respect.
“If I show respect to you, I expect respect as well. That’s just the fact of the matter.”
Ostapenko, 28, is well-known for allowing her frustrations to spill over and has previously received fines for unsportsmanlike conduct.
Afterwards, she posted a message on Instagram accusing Townsend of being “very disrespectful”.
“I told my opponent she didn’t say sorry, but her answer was that she doesn’t have to say sorry at all,” Ostapenko, who won the Roland Garros title in 2018, wrote.
“There are some rules in tennis which most of the players follow and it was the first time ever that this has happened to me on tour.
“If she plays in her homeland it doesn’t mean she can behave and do whatever she wants.”
Townsend said she did not expect to receive an apology from Ostapenko.
“Saying I have no education and no class, I don’t really take that personally, because I know that it’s so far from the truth,” Townsend said.
“If I allow what other people have to say about me affect me in that way, then they win.”
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