The Fifers admitted breaching SPFL rules by refusing to play the game after a home player returned a positive Covid-19 test.
Following a disciplinary hearing on Thursday they were fined £11,000 for their actions, of which £1000 is payable now and £10,000 is suspended.
In a statement, the SPFL say a Clyde player had, immediately before the match, received a positive result from a test taken on the previous Saturday which was relevant to their scheduled match tonight (Thursday).
They added that it was "wholly coincidental" that the result of the Saturday test became available on April 27.
The statement added: "The Clyde player was, entirely properly, immediately withdrawn from the Clyde FC squad for the match on 27 April and sent home.
"East Fife were advised by the match referee and representatives of Scottish football’s joint response group that the match could safely go ahead and that none of their players would have been identified as ‘close contacts’ of a positive case by playing in the match.
"In the light of this advice, the SPFL advised East Fife that, should they not play in the match, their failure to do so would be subject to an SPFL investigation and potential disciplinary action.
"East Fife players decided, unilaterally, not to take part in the match and that decision was supported by the club and by its manager.
"East Fife were charged with failing to fulfil the fixture, and admitted breaching SPFL rules G5 and G53.
"East Fife admitted breaching SPFL Rules and confirmed that, if the match were to be rescheduled, they would play the match."
The rescheduled Clyde v East Fife match will now take place next Thursday, 6 May.
A spokesman for the SPFL said: “Clearly, the circumstances surrounding Tuesday’s match were difficult for all involved.
"However, the advice from members of the joint response group was clear and unequivocal.
"Following such advice, which is informed by frequent discussions with the ESCAG, has been, and remains, crucial to achieving an orderly conclusion to SPFL competitions.
“While the SPFL has sympathy with players who may not have had all relevant information, no league or other football body can accept a situation in which a club fails to fulfil a match in circumstances where all of the medical advice is that the match can safely proceed.
"There has been no recorded case, worldwide, of any on-pitch transmission of Covid-19.
“We would like to thank East Fife for their early admission of the charges. We appreciate the difficult position of the East Fife FC board, faced with a decision having been made by their players.”
East Fife manager Darren Young revealed in the minutes that followed Tuesday night's postponement that some of his squad have pregnant partners who refused to play on safety grounds.
Others were concerned for their employment should they be subject to a quarantine through the track and test system.