The incident occurred when the number 19 bus pulled in at a bus stop near the city's Prado beach at 6.30pm. on Monday, newspaper La Provence reports.
On seeing the woman attempt to board the bus with a friend, the driver requested that she remove the veil from her face in line with France's new laws forbidding full-face coverings.
According to a report in Le Parisien, the driver then announced to passengers that he would not resume the journey until the woman had revealed her face and he was satisfied that her identity matched that of the photo on her travel card.
The woman refused to comply with the request, La Provence said, preferring instead to step back down from the vehicle and wait for the next bus.
A number of young passengers who supported the woman's stance then prevented the driver from closing the door and continuing along his route, calling him a racist.
A skirmish broke out and one of the passengers punched the driver in the face.
Police were called to the scene, but by the time they arrived most witnesses had fled. Officers questioned the women at the centre of the incident, who, though unrepentant, said that she had not intended to create a scene.
France's prohibition of full-face coverings, often dubbed the 'burqa ban', came into force in October last year. Anyone refusing to show their face risks incurring a fine.
Marseilles public transport firm RTM said it was launching an internal investigation into the incident. Speaking to La Provence, RTM chief Karim Zéribi criticized what he called "the driver's inappropriate reaction", explaining that the company's employees had received instructions not to challenge veiled passengers if they were in possession of valid travel cards.