Svitolina had said she would not shake hands with Russian or Belarusian players at Roland Garros because of Russia's invasion of her country.
After her 6-4, 7-6 loss, the ninth-seeded Kasatkina gave Svitolina a thumbs-up, and Svitolina was also seen putting her thumb up. It was unclear whether the gesture was addressed to Kasatkina, who was booed by the crowd.
"Leaving Paris with a very bitter feeling," Kasatkina wrote on Twitter. "All this days, after every match I've played in Paris I always appreciate and thanked crowd for support and being there for the players. But yesterday I was booed for just being respectful on my opponents position not to shake hands.
"Me and Elina showed respect to each other after a tough match but leaving the court like that was the worse part of yesterday. Be better, love each other. Don't spread hate. Try to make this world better. I will love RG no matter what, always and forever. See u next year."
Kasatkina has been one of very few Russian players to speak out against the war, calling it "a full nightmare." On Sunday, she was supported by Russian popular singer Zemfira, who left the country because of her opposition to the invasion.
Svitolina said she was "really thankful for her position that [Kasatkina] took."
"Yeah, she's really brave person to say it publicly, that not so many players did," Svitolina said. "She's a brave one."
Zemfira was placed on a Russian Justice Ministry list of foreign agents in February on the grounds that she supported Ukraine and criticized the invasion.
Geopolitics have been at the center of this year's French Open, with Aryna Sabalenka of Belarus skipping her postmatch news conference twice, saying she felt unsafe after being grilled about the war. Belarus has been used as a staging area by Russia for the war.
Last week, two-time French Open champion Novak Djokovic started a controversy after writing on a camera lens "Kosovo is the heart of Serbia" amid unrest in the region.
Information from Reuters was used in this report.