By SABC Sport
5th June 2023
Knowing Ukrainian players' stance of not shaking hands with Russian or Belarusian opponents at the end of matches, Kasatkina gave Svitolina a thumbs up, which was reciprocated, before walking to her chair.
Some of the fans on Suzanne Lenglen then responded by booing the Russian as she walked off court.
Kasatkina wrote on Twitter: "Leaving Paris with a very bitter feeling.
"All this days, after every match I've played in Paris I always appreciate and thanked the crowd for support and being there for the players. But yesterday I was booed for just being respectful on my opponent's 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."
The world number nine has been the most outspoken Russian or Belarusian player against the invasion of Ukraine, provoking a strong negative reaction in her home country.
Kasatkina is also one of the few Russian athletes to have come out as gay and she earned praise from Svitolina, who said of her anti-war stance: "I'm really thankful for her position that she took.
"She's a really brave person to say it publicly, that not so many players did."
Known as one of the most uncompromising crowds in tennis, the spotlight has been on the Roland Garros patrons this year as sport has mixed with politics.
Svitolina and fellow Ukrainian Marta Kostyuk were also booed after not shaking hands, while Novak Djokovic hit out at fans who jeered him while he took a medical time-out.
Svitolina faced another Russian player in the third round, Anna Blinkova, and next she will take on Belarusian second seed Aryna Sabalenka.
Sabalenka has been criticised by Ukrainian players for not speaking out strongly enough against the war and has refused to do press conferences after her last two matches following tense exchanges with a Ukrainian journalist over her previous support for Belarus president Alexander Lukashenko.