Lord Tebbit, 87, told the Daily Mail: "I find it difficult to accept a sodomite as a member of the clergy who will, for example, be called upon to conduct marriage services. I will struggle to attend if he is officiating."
The Tory peer has been attending services at the cathedral for nine years.
He said he learned from a newsletter in March that Canon Joe was going to become the cathedral's most senior official and is in a civil partnership with Rev Chris Eyden.
He added: "I just don't see how he can decently read the marriage service. I stand by what is written in the Bible, which is the basis on which the cathedral was built. It says the cities of Sodom and Gomorrah were destroyed by God.
"It cannot be a sin to be born homosexual. It is what one is. What I can't accept is the activity. I use the expression sodomite."
While Lord Tebbit disagrees with the cathedral's decision, he said he will continue his financial donations to the church.
His comments have been met with backlash with Ivan Massow, the gay former Tory London Mayoral hopeful, calling for him to be expelled from the Tory Party.
Tory MP Crispin Blunt said: "Lord Tebbit may have a disagreeable surprise at how many cathedrals he has self-excluded himself from."
A spokesman for the Diocese of St Edmundsbury and Ipswich said: "It has been clear for more than a decade that clergy are entitled to be within civil partnerships. Lord Tebbit is welcome to his views."
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