Inquiry chairman Lord Hardie said he had decided it would be unfair to let him take the stand when he was clearly unwell.
And he adjourned the inquiry for a couple of hours until the next witness, Deputy First Minister John Swinney would be available.
Mr Stevenson will return to give evidence when he has recovered.
Lord Hardie also revealed that extra documents relating to the tram project had been discovered during an office swap which meant the appearance of former city council chief executive Dame Sue Bruce and two others who helped negotiate the final deal to complete the line - troubleshooter Colin Smith and former TIE chairman Vic Emery - would be delayed until next month.
Lord Hardie said the council had earlier provided documents relating to Dame Sue’s involvement in the project as requested.
“However on November 10 a solicitor from Edinburgh contacted the solicitor for the inquiry to make him aware that in the course of the current chief executive moving office, further documents had been discovered. We then recovered these - they were substantial and they required to be analysed.
“Clearly it was not appropriate to take the evidence of these three witnesses without these documents and it was necessary in fairness to the core participants that they are given an opportunity of considering them.”
Lord Hardie said after this week the inquiry would adjourn until the week beginning February 19.