Air carrier flight crew reported a ground conflict with a tug during taxi out at IAH airport.

Narrative

After receiving clearance to taxi (SC-RA-RB-Spot X); and area around the aircraft being cleared by Captain (CA) and First Officer (FO) from Spot Y in the Echo alley and the taxi light being illuminated; the CA attempted to begin his taxi when suddenly a rapidly approaching tug from the right appeared. The tug was not slowing its rate of closure with the aircraft and the CA applied manual brakes to stop the aircraft from colliding with the tug. Only after the aircraft stopped did the tug briefly stop and then continued to move around the front of the aircraft. FO notified the Ramp Tower who was more interested in the three aircraft who were holding on nearby taxi ways for us to taxi clear of the alley than the near accident that was nearly avoided. The crew immediately made a PA informing the passengers of the reason for the abrupt stop. The CA spoke with the Flight Attendant's (FA) to check their status and that is when he was informed that one of the forward FA's had bumped her left forearm against the galley structure. When he asked if she needed help; another FA responded with 'She's Okay'. I asked the FA's to perform a wellness check on our passengers as we resumed our taxi and they stated they would. Later the crew was informed that the wellness check had been done; that all passengers had been seated when the incident occurred and the wellness check revealed no injuries to passengers.