Janette Williamson, who is standing for as the Labour candidate for Wirral council, has apologised for tweeting "why the f**k isn't he dead yet?" in a discussion about Prince Philip, as well as for saying that Ed Miliband "needs to grow a set of Ed Balls".
The Twitter account seems to have been deleted, but not before Wirral's Conservative leader Jeff Green saw them and complained to Miliband.
Williamson said: "I would like to make a full and unreserved apology for my comments on Twitter.
"I appreciate the tone and content of the messages was completely inappropriate and would like to say sorry for any offence caused.
"I feel I have let myself and the Labour Party down and promise to learn from my mistake."
The leader of Wirral Borough Council has said in an email to Mr Miliband: "While the remarks about you, your MPs and the party you lead are of course a matter for you, I believe her comments on HRH The Duke of Edinburgh to be totally unacceptable and incredibly offensive.
"I am therefore asking you, as leader, to disassociate your party from her remarks and for you to consider whether this person is the acceptable face of the Labour Party in Wirral at the elections on May 3."