On Wednesday House Speaker Nancy Pelosi withdrew an invitation for him to address Congress, saying government services should fully reopen first.
Mr Trump initially said the speech should be "on time, on schedule, and very importantly, on location."
But in a tweet overnight he conceded he could not deliver on his pledge.
He acknowledged Ms Pelosi's refusal to allow Congress to host the State of the Union address until the shutdown - now in its 34th day - had been resolved.
"This is her prerogative," he said.
Ms Pelosi responded with her own tweet saying she hoped Mr Trump would back a bill passed by the House of Representatives to fund the agencies affected by the shutdown.
"Please accept this proposal so we can re-open government, repay our federal workers and then negotiate our differences," she said.
Democrats have argued that with so many federal employees furloughed - temporarily laid off - or working without pay, the high-profile State of the Union address, which involves both chambers of Congress and the president, would not be logistically feasible.
But Mr Trump has denied that there were any security concerns posed by the ongoing partial shutdown.
The Speaker, who is the third most powerful politician in the US, had invited Mr Trump at the start of January to address Congress on the agreed-upon date of 29 January.
"At that time, there was no thought that the government would still be shut down," Mrs Pelosi said in her letter on Wednesday.
"I am writing to inform you that the House of Representatives will not consider a concurrent resolution authorising the President's State of the Union address in the House Chamber until government has opened."
"Again, I look forward to welcoming you to the House on a mutually agreeable date for this address when government has been opened."
Mrs Pelosi had first asked for the State of the Union, a keynote agenda-setting speech by the president, to be delayed on 16 January, citing an unfunded Secret Service and Department of Homeland Security.
In retaliation, Mr Trump last week denied Mrs Pelosi use of military airplanes on a trip to Brussels and Afghanistan - less than an hour before her flight was due to depart.
He asked her to stay to negotiate an end to the partial US government shutdown.
A row between the Republican president and Democrats over border security has led to the longest government shutdown in US history.
Mr Trump is demanding $5.7bn (£4.4bn) of congressional funding to build a wall on the US-Mexico border, but newly empowered Democrats have refused.
Some 800,000 federal employees have been going unpaid since 22 December as a result of the shutdown.