The match at the Camp Nou ended 2-2, with Antoine Griezmann only being brought on for the final couple of minutes by Barca head coach Quique Setien.
The treatment of Griezmann led to the outburst from Dugarry, who thinks his compatriot should strike Messi to try and sort out the alleged ill-feeling between the pair.
"What is [Griezmann] afraid of? A kid who is 1.5 metres tall and half autistic?" he said on RMC Sport.
"All he has to do is show some balls at some point.
"I've been saying for a year that he has a problem with Messi. He has to hit him in the face.
"It's true that Messi could give him more passes, but honestly I am not surprised. Griezmann loses balls, doesn't play with confidence. Griezmann should go and talk to Messi to solve the problem."
The story took a further twist as Dugarry took to his social media page to apologise for his offensive words, only to later delete the post.
"I’m sincerely sorry for the breadth and depth of my remarks about Lionel Messi," he originally tweeted.
"I didn't want to stigmatise people with autism disorders, that was not my intention. I apologise to the people I shocked and will do it again (apologise) on Team Duga (TV show) this evening."
Griezmann has struggled for form since making his big money move from Barcelona, scoring just eight league goals in La Liga, earning a 84 FC Rating for his performances.