I hate to burst your bubble, but the gulf stream has shut down before. It caused a relatively short ice age. I say relatively because I believe it was in the hundreds or thousands of years, long for us, short compared to some ice ages. I don’t think the Earth has ever stopped spinning, though.

The spin of the Earth does cause currents to flow, but its not the only factor. Cold water sinking in the north by Greenland plays a major role in where the gulf stream goes. The pressure principle you've been talking about, its applied right there. The cold, sinking water causes a pressure difference. As it sinks, it pulls warm water up from the southern latitudes. You don’t think this might change if its no longer pulling warm water, but much cooler water?