It had been widely anticipated Mr Heaton-Harris would announce the date for an election on Friday, after a deadline to restore devolved government at Stormont passed at midnight.
Instead he said he would give more information next week and would meet with the Stormont parties.
He said he was faced with “limited options”.
Speaking in Belfast on Friday, Mr Heaton-Harris said: “I am deeply disappointed we are where are are now.
“This is a really serious situation. As of a minute past midnight last night, there are no longer ministers in office in the Northern Ireland Executive.
“I will take limited, but necessary steps to ensure public services do continue and to protect the public finances. But there is a limit to what the Secretary of State can do in these circumstances.”
With no ministerial executive in place, the UK Government assumes a legal responsibility to call another election.
Stormont ministers, who have been operating in shadow form since the Assembly collapsed earlier this year, also ceased to hold office at midnight.
Responsibility for running devolved departments will now pass to senior civil servants, although their powers are limited.
During a visit to Thurrock, in Essex, earlier on Friday, Labour leader Sir Keir Starmer told reporters: “We do need to get the institutions in Northern Ireland back up and running and the sooner we can do that, the better.
“That requires our Government to get people around the negotiating table and to resolve the issues on the protocol.
“I think they can be resolved. I think there are amendments that can be made to the protocol without ripping it up.
“But the Government needs to get people around the table and get that agreement because at the moment we’ve got a very difficult situation for people in Northern Ireland who are suffering as a result of the government’s failure to convene this.”
The DUP is blocking the restoration of powersharing as part of its protest against the post-Brexit Northern Ireland Protocol.
Environment minister Therese Coffey said “clearly” there was not “sufficient agreement” among Stormont parties to avoid a fresh election.
“I think it’s regrettable the parties were not able to come together to form that executive,” Ms Coffey told Sky News.
“But the law was clear. We passed the legislation that this would happen and clearly there wasn’t sufficient agreement to be able to avoid the elections."
But DUP leader Sir Jeffrey Donaldson insisted it was the fault of the UK Government that devolved government in Northern Ireland was not functioning.
“We’ve had six months in which to do something about the protocol, and during those six months we have had three prime ministers, we have had the Government changed often and we haven’t seen the progress that is needed,” he told BBC Radio Ulster.
“I think the Government would be within its rights to say, given that those six months have elapsed and progress hasn’t been made, that we need a further period to sort this out, get a solution on the protocol that restores Northern Ireland’s place within the UK internal market and that will see the institutions restored immediately.”
Sinn Fein MLA Conor Murphy dismissed the notion the DUP tactic of refusing to form an executive was exerting any pressure on the Westminster Government to act over the protocol.