That was followed later by Kilmarnock's all-Scottish Premier League clash with Falkirk, which was postponed over safety concerns.
The call-offs were the first in what is expected to be a decimated fourth-round fixture card this weekend as the cold snap continues to wreak havoc with the football programme.
The Scottish Football Association have confirmed there will be pitch inspections for six of the remaining 14 ties on Thursday, all of which are being hosted by Scottish Football League clubs with no undersoil heating.
They are Raith Rovers v Airdrie, Ayr United v Brechin (both 9.15am), Forfar v St Johnstone (10am), Albion Rovers v Stirling Albion, Edinburgh City v Montrose, and Ross County v Inverurie Loco Works (1pm).
None of them are expected to survive.
The loss of the Morton v Celtic match came as no surprise after the Irn-Bru First Division club took the SFA's advice and held a pitch inspection as early as possible.
Morton said in a statement: "The pitch inspection was held earlier than normal due to the current extreme weather conditions, and in the interests of supporter safety and convenience.
"Given the weather forecast for the week ahead, it was felt that it was essential for the inspection to be carried out as quickly as possible."
The fixture has been rescheduled for Tuesday night (kick-off 7.45pm) with another inspection planned on Monday.
But, unless there is a significant thaw, it looks set to be postponed once again until the following Monday.
Kilmarnock v Falkirk has also been rearranged for Tuesday night, with Strathclyde Police expected to make a decision as early as Sunday as to whether the game can go ahead.
As with Killie's postponed SPL game with Celtic last week, the Rugby Park pitch is playable but concerns lie with access roads to the stadium.
Kilmarnock said in a statement: "With ongoing low temperatures forecast and the possibility of further snowfalls, the decision has been taken to postpone the match at an early date in order to reduce inconvenience to supporters of both clubs.
"A further meeting with Strathclyde Police will take place on Sunday in order to assess the possibility of the match proceeding on Tuesday."
Press Association Sport also understands Hamilton are not confident of Sunday's tie with Rangers taking place for similar reasons, with an inspection planned at New Douglas Park on Friday.
Aberdeen will also hold a Friday inspection ahead of Saturday's all-SPL tie with Hearts.
A Pittodrie spokesman said: "At present, the pitch is fine and we're preparing for Saturday afternoon as scheduled.
"However, we'll be monitoring the situation closely over the next 48 hours and doing all we possibly can to ensure the game goes ahead with a final decision likely to be made on Friday to avoid supporters making unnecessary journeys."
Safety fears mean all ties are potential casualties but those hosted by SPL clubs could beat the weather due to undersoil heating.
That also applies to two matches held at SFL grounds, Inverness v Motherwell and Partick Thistle v Dundee United.
Friday is the likely day for pitch inspections at these ties.