St Helen's Bishopsgate said a successor to former Bishop of London Richard Chartres who supports same-sex marriage would be met with "unavoidable avoidance".
Rector William Taylor told his congregation his first question to the next Bishop of London, who is yet to be named, would be whether he's willing "declare as sin what God calls sin".
The leader of the St Helen's, which regularly attracts 2,000 to its four Sunday services, said "avoidance" of believers who accept "what the Bible calls sexual immorality" is legitimate.
In a sermon, Rector William Taylor endorsed the stance Richard Chartres took on the issue, adding: "We now wait to see what the new Bishop of London's views are.
"My first question to him or her will be: 'Are you prepared to openly to declare as sin what God calls sin and to summon all people to repentance and to do so publicly?'
"If the answer is 'no' then there is an 'unavoidable avoidance' for us all."
After final interviews for the new Bishop of London concluded last week, Downing Street will be asked to approve any name put forward.
Premier has approached St Helen's Bishopsgate for a comment.