The Rev Canon Dr Gavin Ashenden announced he would resign after the Queen's intervened over his outspoken opposition to the issue.
It comes after he criticised the decision by Rev Kelvin Holdsworth, Provost of St Mary's Cathedral, Glasgow, for inviting a reading from the Islamic holy book at the Epiphany service on 6 January.
The recitation included verses which explicitly denied that Jesus was God's son – a key Christian doctrine.
Ashenden called for Holdsworth to apologise and hinted he should quit his post.
"Some people will wonder if he has sufficient theological astuteness to remain in such a responsible office in times as fraught as these?" he wrote in a blog post. "Our interfaith relations require both integrity and competence. The strategy in Glasgow appears to have been short of both."
Announcing his resignation after nine years as one of the Queen's 34 chaplains, he said he realised "a very important convention that the Queen should not be drawn into publics affairs where she is deemed to be taking a position".
He added it was often wrongly assumed that he, as chaplain, had the Queen's confidence.
"It does the Queen no good at all for it to be assumed by any of the public, or the fourth estate, that she does have a view that is being expressed by someone connected with her," he wrote.
"That being the case, I could most easily avoid any misunderstanding by not speaking out in the public space on matters of faith that I took to be important. This would have the effect of silencing me and prohibiting me taking part in public debate.
"On the other hand, if I did choose to speak out, as a matter of integrity and responsibility, I ought not to do it while I was in possession of the office of 'Chaplain to the Queen'.
"Because I think it a higher and more compelling duty to speak out on behalf of the faith, than to retain a public honour which precludes me doing so at this time, I resigned my post."