That's not what he said. John Kerry said:
But here is a fundamental reality: if the choice is one state, Israel can either be Jewish or democratic – it cannot be both –and it won’t ever really be at peace.
So what he is trying to say here is that
In his opinion, Israel should embrace a two state solution.
If Israel does not split off Palestine into a separate state, it has to choose between being a Jewish state that does not represent the Palestinian Muslims. Or, it can be a democratic state that is majority Muslim.
He is not saying that countries in general can't be both religious and democratic. His criticism is specific to Israel as a single state that includes the Palestinian areas. In fact, under his idealized two state solution, presumably both states would be religious and democratic. Israel would be Jewish and Palestine would be Muslim.
In a way, this is an offering to Israel. If they surrender to his terms, he believes they could be Jewish and democratic in their portion of the current country. However, many people aren't reading it that way.
There have been a number of criticisms of this stance. For example, it's not clear that the two states would achieve peace. This doesn't resolve the overall problem that most Middle Eastern Muslim countries do not support Israel's right to existence as a Jewish state, at least not popularly (the leadership may).
Note that such criticisms have their own detractors. Obviously Kerry doesn't agree with this, and he's not unique in that.