Pat Dobson is correct, but the answer is not complete. Peter Gabriel himself has remarked on this very song:
“I uncovered a lot,” says Gabriel, “and it’s in the songs and on the new album, like ‘In Your Eyes.’ On two recent trips to Senegal, it was explained to me that many of their love songs are left ambiguous so that they could refer to the love between man and woman or the love between man and God. That interested me, because in our society it’s a little like the sacred versus the profane—you know, church music, for instance, expresses a religious type of love, and romantic love belongs to the Devil, if you like.”
“So I began playing in the lyric with a mixture of the two:
“In your eyes
I see the doorway to a thousand churches
In your eyes
The resolution of all the fruitless searches
In your eyes
I see the light and the heat
In your eyes
Oh, I want to be that complete”
The "search" must mean the search for love, and the church is the resolution of that search. A thousand searches opens up the possibility for a thousand churches, i.e. a thousand resolutions. The resolution, i.e. final resolve, of a search for love is marriage (in popular conception, especially then, but even now). So those churches represented the possibilities for different marriages, until the right one was found.
But the reason he chose the church as the symbol, as opposed to any other, is that it further adds to the ambiguity. The words could just as easily be taken to mean the love between a devotee and their god. The other churches are other religious paths, but in the end those were fruitless endeavors.
Christianity's long history of marrying religious devotion and matrimony further adds to the success of the ambiguity, even down to the very same word we use in English: love.