Will a warmer climate stop the Gulf Stream?

The Gulf Stream brings the warm water from the Mid Atlantic northward along the west coast of Scandinavia's and makes our climate much warmer than normal in our latitudes than in other parts of the world.

In the Arctic Ocean and the seas east of Greenland seawater sinks from the surface down to deeper levels due to high density.

The question is whether theories such as in "The Day After Tomorrow" are true? It claims that increased temperature through global warming, could lead to an increased supply of fresh water in the Arctic sea from the glaciers. The salinity in the water would decrease and the density would become lower. But how much depends on the salt and how much is due to the water's properties? Water is at its heaviest at 4 degrees and a high salinity also increases density. The Deep water formation, when the salt, heavy surface water sinks and is replaced by inflowing, warmer water from the North Atlantic is the main engine behind the Gulf Stream.

Can deep water formation be slowed down or completely stop if the surface water density decreases due to salinity? The flow of warm water from the Atlantic would in such cases subside or stop. Should the Gulf Stream come to a halt, would obviously be catastrophic. The question is how likely this is?