Visitors India are set to play their last Test of the five-match series against England from Thursday at Surrey County Cricket Club's stadium, with the hosts leading 2-1.
Gambhir, a 58-Test veteran, was caught on camera wagging his finger as Fortis towered over him, repeatedly stating: "You can't tell us what to do" and "You don't tell any of us what to do".
Gambhir can also be heard adding: "You're just the groundsman, nothing beyond."
Fortis is understood to have been keeping a protective eye on the square during India's practice with plenty of cricket still to be played on it once the international summer is over. Gambhir did not take kindly to his intervention.
After walking away in conversation with India batting coach Sitanshu Kotak, who appeared to take on the role of mediator, Fortis told Gambhir he may be forced to lodge a complaint against him, drawing a response of: "You can go and report to whoever you want".
Speaking later to the press, India batting coach Sitanshu Kotak told reporters Fortis' objections to a coolbox being wheeled on to the square kicked off the spat.
"He was sitting on the roller, he shouted and told the support staff not to take it there," he said.
"It is good to be a little possessive and it is also good to be protective, but not that much. Gautam just said do not talk to the support staff like this.
"It's a cricket wicket, there's a five-day Test starting the day after, and we were wearing joggers. So it was a bit awkward.
"We all know that curators are a little over-protective ... but at the end of the day it's a cricket pitch, not an antique that you can't touch."
Fortis downplayed the incident when he spoke to a group of journalists.
"There's nothing more to add ... we have nothing to hide here," he said.
Surrey did not immediately respond to an email requesting comment.
Kotak said the India players and staff had taken care not to cause any damage to the playing surface during their practice session earlier on Tuesday.
Heated verbal exchanges in the last two Tests have added to the tension between the teams in a competitive series.
England did not train, preferring to let their players rest with three days between the Tests.
with PA
