The 44-year-old handler wasn't impressed by the referee either, but it was McDowell who drew the brunt of the Australian's ire after he ruled that Allister Hogg had stepped into touch when he rampaged through the French defence to score a try.
That would have put the Scots at least 15-6 to the good with 10 minutes remaining, but instead they saw that turned round as Jon Petrie was sin-binned and then, having drawn level through a Yann Deliague drop goal, the hosts stole it when Hugo Southwell's kick was charged down and Damien Traille ran onto touch the ball down.
"Three outrageous decisions by the same official in three different matches and they keep on appointing him," stormed Williams, who has only two test victories to his name since taking over from Ian McGeechan in November 2003.
"He refused to refer the Italian try to the video referee last year to see if it had been properly grounded. Then against Australia last autumn he said the Wallabies had not knocked it on when in fact they had which led to a try that effectively cost us the match and now he claims that Allister Hogg's foot was in touch.
"He wasn't even down to officiate at this match and he suddenly turns up out of the blue without us being informed. Three times the same guy and same problem ... do we have a problem with him? Yes."
Williams, though, saw much to be proud of in the performance by last year's wooden spoonists, who have also had a traumatic time off the pitch with several board members of the Scottish Rugby Union resigning over proposed reforms to the game.
"I want to point out the tremendous effort of this Scottish team in a period of incredible negativity. They have taken it on and shown leadership. They had to perform for the nation and it was a magnificent performance. We deserved victory and to my mind we gained that victory," added the former Leinster coach.
The French manager Jo Maso wasn't inclined to disagree. "It was a hold-up. I really didn't think we were going to win as we had little idea of how to score. However, we got a drop goal to draw level and then the try came from a mistake by one of their players (Hugo Southwell had a kick charged down and Damien Traille touched down).
"We will go away and analyse what went on and we will act on it before the England match next Sunday," added Maso.
France coach Bernard Laporte said his players had not stepped up to the plate in the first-half which they ended 6-0 behind thanks to two Chris Paterson penalties.
"They played like frightened players in the first-half as if they didn't want the ball and weren't prepared to take risks," said Laporte, who had demanded that his new halfback partnership the 23rd different pairing of his 60 test reign of Pierre Mignoni and Yann Delaigue act as leaders.
"I told them at half-time to put their foot down and up the tempo and they did to a certain extent. There are some excuses, such as the back row were new to each other and both Sebastien Chabal and Julien Bonnaire were more evident in the second-half. But the first-half was not an impressive sight," added Laporte, who might have noted that the dropped Frederic Michalak received the biggest cheer when the players names were called out and on his entry into the match late in the second-half.
Laporte admitted to being a little perplexed by the fearful way his players had performed, which saw them knock-on 21 times and lose six of their lineouts with captain Fabien Pelous looking especially vulnerable.
"There were no such indications during the week that they were nervous," he said. "I guess some felt the pressure of returning to international rugby while others knew they were playing for their places.
"However this reminds me a bit of our match against Italy here in 2002 where we didn't play that well but went on to win the Grand Slam," he added.
FRANCE: Elhorga, Rougerie, Traille, Liebenberg, Dominici, Delaigue, Mignoni, Marconnet, Servat, De Villiers, Pelous, Thion, Bonnaire, Tabacco, Chabal, Valbon.
Replacements: Valbon for Rougerie (18), Michalak for Delaigue (74), Bruno for Servat (80), Milloud for De Villiers (52), Lamboley for Thion (70), Nyanga for Tabacco (64), Yachvili for Valbon (80).
SCOTLAND: Paterson, Danielli, Craig, Southwell, Lamont, Parks, Cusiter, Smith, Bulloch, Kerr, Grimes, Murray, White, Hogg, Petrie.
Replacements: Douglas for Kerr (55), Hines for Murray (66), Dunbar for Petrie (80).
Ref: N Williams (Wales).