Craig Melville is said to have sent a number of messages to an unnamed woman a day after the Paris terror attacks.
He has already been suspended from his role as an aide to SNP deputy leader Stewart Hosie.
Dundee City Council confirmed that its chief executive had received Mr Melville's letter of resignation.
In a statement issued to the Courier, Mr Melville said: "The pressures of the recent deeply distressing allegations are having an adverse effect on the health of my family and I.
"Therefore I do not feel that I am currently able to fulfil my duties as councillor for the aforementioned ward and to continue as convener of the environment committee to the best of my abilities."
In one message, Mr Melville, who was the city's environment convener, is reported to have referred to the SNP's international development minister, Humza Yousaf.
A Dundee City Council spokesman said: "We can confirm that a letter of resignation from Mr Melville has been received by the chief executive.
"A by-election will be called for the Maryfield ward and arrangements will be made in due course."

It comes after Aberavon MP Stephen Kinnock said the poll result could cause "irreparable damage" to the industry.
Newport-based steel group Liberty is determined to press ahead with its bid, BBC Wales understands.
Tata said no interested parties had changed their positions.
A spokesman added: "Like the rest of the business world we will be taking the time to consider the implications of the verdict of the British people.
"We have received no notifications from any of the interested parties of any change in their position."
Concerns had been raised in the run-up to Thursday's referendum that international buyers for Tata Steel's UK business would drop out if the result was a loss of access to European markets.
Mr Kinnock told BBC Radio Wales he feared for the Port Talbot plant amid rumours bidders were pulling out after the vote to leave the EU.
"I spent almost the entire referendum campaign saying 'if you want to save steel industry, you must vote to remain'," he said.
"I find it absolutely extraordinary that that hasn't been understood because of the uncertainty it now unleashes.
"The vote has taken place, the people have spoken, but I am deeply, deeply concerned about the future of our steelworks in the light of this vote."
Asked about the future, he told the Sunday Supplement programme: "I'm already hearing rumours that the bidders are going to now pull out.
"If you look at the government, it was actually the PM that was pushing this forward," he said.
"The prime minister realised that it was a political priority - he's gone now, so where is the voice at the top level of the British government standing up for the steel industry? I don't see it.
"The enemy of business is uncertainty and that is fundamentally why this referendum vote is going to potentially cause irreparable damage to the steel industry."
First Minister Carwyn Jones told the programme he was "still fighting" to keep the Port Talbot plant open.
"I'm not going to pretend to you that Thursday helped because it didn't," he said.
"I think if we'd seen things different on Thursday we'd have been in a stronger position with regard to Tata.
"But now things are once again up in the air."
Welsh Conservative leader Andrew RT Davies, on the winning side of the referendum result, called the first minister's comments "scare-mongering of the highest order".
"If we get this [withdrawal from the EU] right, there are enormous opportunities," he said.
"The deals and the trade agreements are in place, business continues."
A Dundee councillor currently under investigation for allegedly sending anti-Islamic texts to a Muslim colleague has resigned.
[NEXT_CONCEPT]
No bidders have pulled out of trying to buy Tata Steel's UK operations in the light of the Brexit result, the firm has said.