The expressive message of the church is that the word of God is infallible, totally correct, and was translated entirely according to His will. In the word, there are scriptures that call homosexual behavior an abomination, sin, etc. Therefore, condoning such sin by allowing/performing gay marriages alters their expressed message.

That is as simple as it gets.

Beyond that, the lack of any alteration would not result in private organizations changing how they operate because, again, those private organizations have the right to determine what they will or will not do.

The government cannot force churches to perform gay marriages.