In some capacity, differential heating is responsible for convective motion of the ocean due to buoyant forces. However, it is possible to show that the net (vertically integrated) horizontal transport in most of the ocean is solely due to the action of the wind stress on the water surface (Sverdrup, 1947). In other words, the pattern of major ocean currents is a direct mirror of the pattern of the wind stress.
The main reason the current is intensified in the western boundary is because of the $\beta$-effect, or the strengthening of Coriolis with latitude. This effect causes the slope of the ocean surface to be steeper on the western side because the equator easterlies are stronger than the mid-latitude westerlies and therefore surface water piles up in the western boundary. Due to conservation of potential vorticity the steeper slope results in faster geostrophic flow on that side of the gyre (i.e. western intensification).
Another cause for the westward intensification are the density gradients, namely the sign of the gradient of zonal density. For example, if the density in the polar regions suddenly became less than the density in the equator regions, the Western boundary current would be considerably weaker. Another argument explains the phenomenon of western intensification via the accumulation of energy due to westward propagating planetary Rossby waves.
If we assume that the current velocity in the Gulf stream is $U \sim 1$ m/s on average (its maximum may be twice that) then the momentum per unit volume of a fluid parcel will be $\rho U \sim 1000$ kg m$^{-2}$ s$^{-1}$. Typical estimates suggest the total mass transport across a cross section of the Gulf stream is on the order of $90 \cdot 10^6$ m$^{3}$ s$^{-1}$ of water. By comparison, the momentum of a fluid element in the jet stream in the atmosphere is one order of magnitude smaller than that in the Gulf stream, despite the fact that the wind speed is one hundred times larger than the water current speed.
Assume the gulf stream has a total volume $V = L \cdot W \cdot D$, where the length $L = 2500$ km, the width $W = 75$ km and the depth $D = 800$ m then, as a crude approximation, we can estimate the total force $F$ that must act on the fluid over (lets say) $2$ years to cause a sufficient change in the bulk fluid momentum $m \Delta v = \rho V \Delta v$ so that the Gulf Stream comes to a stop. From Newton's Second Law and plugging in some typical values:
$F = m \frac{\Delta v}{\Delta t} = 2.5 \cdot 10^9$N
Suppose the force is applied through a surface wind and suppose its action extends all the way to the bottom of the current. Then one would require a wind stress $\tau = 0.01$ Pa acting for at least two years, of course that number would need to be higher since in reality the wind stress accelerates the top layer of the water column, and its effect weakens with depth. Note that typical oceanic values of wind stresses are about $0.1$ Pa. This indicates that, purely from physical principles, stopping the Gulf stream within two years is realistic.
However, it is not possible without a complete reversal of well established wind patterns combined with a drastic reduction in the density near the poles compared to near the equator. The latter is why some models predict future slowdown of the Gulf Stream, as the anthropogenic heating of the earth causes the melting of polar ice, the ocean water becomes fresher and less dense. Fortunately, as @naught101 pointed out, changes of that magnitude are unlikely to take place in the near future.
Sverdrup, Harald Ulrich. "Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific." Proceedings of the National Academy of Sciences of the United States of America 33.11 (1947): 318.
Pedlosky, Joseph. "Geophysical fluid dynamics." New York and Berlin, Springer-Verlag, 1982. 636 p. 1 (1982).
