The 30-year-old Englishman was shown two yellow cards in the space of five minutes at the Stade Velodrome before an Andre Ayew own goal gifted victory to the lowly visitors.
Barton told the club's website, www.om.net: "I am very disappointed. I am partly responsible for this loss.
"I have to apologise to my team-mates. The referee made his decisions, but I don't think either of my tackles were too dangerous."
OM coach Elie Baup admitted the dismissal lifted Nancy, who moved off the bottom of Ligue 1 courtesy of the three points - but maintained that Barton had been slightly unlucky.
Baup said: "We have just taken a big slap. We decided to play with a defensive system and it had been working.
"The game turned with the expulsion of Barton. We had not been brilliant in the game, but there was no real danger (before the red card). Being down to 10 men made things more difficult.
"His first card was unjustified since there was no foul, and on the second he was committed to trying to get the ball.
"We waited a long time and wondered when Barton would be sent off for the first time, and that was done tonight. But he did not deserve to be sent off."
Marseille remain six points behind leaders Paris St Germain.