Councillor Alexander made the offensive remarks in a cabinet meeting on Tuesday last week during a debate on health service provisions for people living in Powys. She has since apologised and said she had made an "error in judgement".
In her resignation letter she wrote: "This matter appears to be feeding the toxic nature of division within the county council, it has been made clear to me that I cannot remain on the cabinet,
"I am not a racist and I believe that by resigning, I am accepting a slur on my character which is entirely untrue.
"My resignation has not been tendered without pressure and I would have expected to encounter more loyalty.
"It is, alas, a situation which has to be resolved and you are of the opinion that I should resign."
A spokesman for the council said Ms Alexander would remain a county councillor, representing the Banwy ward.
Councillor Alexander told last week's meeting she and a constituent called Meirion had both been told while trying to access health services in England that their names were just "a meaningless jumble of letters".
She went on to ask fellow councillors what would have happened if her constituent's name had been "Ranjit" and he had been told the same thing. She said it would have been all over the papers.
She added: "We are treated like n*****s over the border. Our language and culture is trampled on and it's a very sensitive issue. It's not our fault that we don't have a district general hospital to go to."
Councillor Alexander said on another occasion she had asked in hospital if a Welsh speaking nurse could say good morning to her father and she was told "that's fine as long as you don't mind your father being considered to be racist".
Fellow councillor Stephen Hayes, who covers Montgomery, had earlier said there should be a vote of no confidence brought against Councillor Alexander if she refused to step down.
Councillor Hayes said: "Following the extraordinary and offensive comments made by Councillor Alexander in last Tuesday's Powys County Council cabinet meeting, I know I speak for many Powys county councillors, of all political persuasions, in deploring her remarks and wanting to dissociate myself from them most emphatically.
"Her comments, made in a meeting open to the public, show a desperate failure to understand what is appropriate behaviour and language for someone holding public office. Her defence that the comments were not intended to be racist is barely plausible, given the context in which they were made."
Top councillor in racist word storm
Powys councillor sorry for error of judgement