Strangely, it might affect France more (at first). Gustav III had met Louis XVI and Antoinette (according to Nagel, he'd even joked about formalizing the Franco-Swedish alliance by wedding Gustav IV to Madame Royal - it's unlikely, but it could be interesting to her wedding another Bourbon which was the only considerations during her parents' lifetime). He was personal friends with Count Fersen. I could definitely see him doing more to get Louis XVI/Antoinette out of France than what either Regent Carl or Antoinette's own family did.
That said, the Flight to Varennes was in June 1791, Gustav was murdered in March 1792. Fersen tried to enlist various monarchs' aid for Louis/Antoinette thereafter, but nobody was willing to get involved AFAIK. Leopold II died two weeks before Gustav was murdered, and Franz II had never met his aunt, so was more interested in making peace with the French govt. I could see Gustav attempting to use his connections to Prussia (his mother was a Hohenzollern) or even Britain (his wife was half-British) to try and get Louis/Antoinette released. Or even just Mme Élisabeth, Antoinette and her kids (à la Henriette Maria and Charles II)