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."
A Dundee councillor currently under investigation for allegedly sending anti-Islamic texts to a Muslim colleague has resigned.