Hayes will replace Scotland's Euan Murray, who damaged his ankle early on during Tuesday's 20-8 tour win over the Southern Kings.
Commenting on the call-up, McGeechan said: "John Hayes has been called up to replace Euan Murray who sustained an ankle injury in the match against the Southern Kings in Port Elizabeth.
"The medical team have now fully assessed his injury and have determined that the injury is serious enough to mean he will play no further part on the tour.
"Euan left the field because of the injury and has damaged the structures around the ankle joint. He is subject to further assessment, treatment and specialist opinion but may require surgical repair.
"John Hayes will be a straight swap for Euan in the squad."
He added: "John is an experienced international front row forward who has played 94 times for his country. He was an integral part of the Irish Grand Slam win this year.
"He also played four provincial matches on the 2005 tour to New Zealand and won a Lions Test cap against Argentina prior to the tour."
It is anticipated that Hayes will fly to South Africa over the weekend and will meet up with the squad in Durban.
Meanwhile, James Hook continues to progress through the return to play protocol for a player returning from concussion.
The Wales back is under continual monitoring by the Lions medical team. Any decision on the date for his return to play will be made in due course.