Ivory Coast international Bailly, 28, who is on loan from Manchester United, was red-carded after just 15 minutes of the tie at fourth division amateur side Hyeres when his foot crunched into the chest of Almike Moussa N'Diaye.
The stricken Hyeres player was stretchered off before being taken to a local hospital from the modest 8,000-capacity Francis-Turcan stadium in Martigues in southern France.
Hyeres president Mourad Boudjellal, the colourful and controversial former owner of Toulon rugby club, told AFP that N'Diaye was then transported to another hospital in Marseille, 40km away.
"He was transferred to Hopital Nord in Marseille and they admitted him to intensive care, as a precaution," said Boudjellal.
"We pray that everything goes well for him. He's a great lad and one of our best players."
Earlier, coaching staff at Hyeres had been more reassuring.
"It's more fear than harm, nothing serious, maybe a concussion," assistant coach Zaki Noubir told Maritima radio.
"Bailly came quickly into our dressing room to offer his shirt and above all to hear the latest news."
Despite playing most of the last-64 tie with 10 men, Marseille still had too much firepower and ran out 2-0 winners.
"On the red card, I didn't see what happened," said Marseille coach Igor Tudor.
"But the sending off changed everything and we had to push hard to win. It was a typically dangerous French Cup game.
"We'll see later what the penalty will be for Bailly."
Alexis Sanchez opened the scoring with a 45th-minute 'panenka' penalty. Bamba Dieng added the second in the 71st minute.
Second division Rodez, from south-central France, pulled off a shock by defeating Monaco who are fifth in the top flight Ligue 1.
After the tie ended 2-2, Rodez came out on top in the penalty shoot-out 5-4 when goalkeeper Sebastien Cibois saved from Soungoutou Magassa.
Ligue 1 Clermont were knocked out by sixth-division Olympique Strasbourg who claimed a shoot-out 4-3 after the tie ended 0-0.
Last year's runners-up Nice slumped to a 1-0 loss at third-tier Puy. Mohamed Ben Fredj scored the only goal after three minutes.
On Friday, Paris Saint-Germain rested their biggest stars but still won 3-1 at third-tier Chateauroux.
With Lionel Messi, Kylian Mbappe and Neymar not even on the visiting bench, PSG took the lead after 13 minutes with a right-foot shot from Hugo Ekitike.
The home team hit back with a 37th minute goal by Natanael Ntolla.
Carlos Soler put the visitors back in the lead in the 78th minute with another Spaniard, Juan Bernat, scoring the third goal in added time.
