It is not a question that can be answered by thermodynamics alone. It is like asking - suppose you have a gas contained on one side of a container by a wall, will the gas eventually expand to the other side of the wall? Clearly the entropy would increase if the gas were to expand so this process is favored by the second law. But you have the wall in the way, so common experience says that the gas will not expand to the other side. But then of course you have quantum tunneling which allows for a particle to pass through the wall with an extremely tiny probability. This means that after an extremely long time you would find the gas to have expanded. For practical purposes, though, you could say the gas never expands.
Now when it comes to this fusion buisiness I think it is clear that the entropy will increase upon fusion. What people seem to be forgetting is that the second law deals with the entropy of the universe. This is why processes like freezing can be spontaneous. If you have some system at constant pressure in thermal contact with its surroundings then the entropy change that occurs in the surroundings due to processes in the system is equal to minus the entalpy of the process. Obviously nuclear fusion of two hydrogen atoms is a highly exothermic process with a massive negative enthalpy value.
So the question is- is fusion possible at all at room temperature? Thermodynamics is silent on this issue. Thermodynamics only distinguishes between allowable and non-allowable processes without saying which of the allowable processes will actually occur or at what rate. Nuclear physics really is ouside my expertise, and I don't know whether it is really possible for nuclear fusion to occur at room temperatures. I would venture to guess, though, that is possible, but extraordinarily unlikely. I guess this because the Boltzmann distribution tells us how energy state are distributed in a system in equillibrium. Hypothetically there are some particles in every state if the system is large enough and the temperature is not 0. This means that there are gas molecules at room T with the same kinetic energy as the average particles on the sun. So hypothetically, there is an outside chance of a collision between two such molecules that results in fusion. If this is true, then thermodynamics says that the rate of the fusion reation will exceed the rate of the reverse reaction up until there is a very high concentration of fused nuclie, so that eventually you would have very little hydrogen left. But we are talking about astronomical time scales here- there is still a lot of hydrogen left in the sun and it is at a much higher temperature and haas been around for billions of years. Since most of the hydrogen that is around now has probably been at higher than room temperature for most of its existence, I think the time scale we are talking about is longer than the age of the universe itself!