Bit of an urban legend apparently. Despite the gulf stream transporting warm water it is nowhere near enough to explain the difference. Richard Seager et al quantitatively put its contribution at around 30% of the difference. The bulk cause is actually air currents and the different topography between Europe and N America. In other words, if there were no gulf stream the ‘warm Europe’ phenomena would still be very much noticeable.

Sources:

academia.edu/…/Is_the_Gulf_Stream_responsible_for… (free)

www.jstor.org/stable/27858802?read-now=1 (free after registration)

Discussion here: …stackexchange.com/…/why-is-europe-warmer-than-no…