The President's powers are quite limited, though they have grown beyond those provided by the Constitution. This is not unique to the US but is part of the trend of growth of executive power in representative governments since the early 1900s. More regulations having the force of law are made by the various agencies of the executive than by the legislative process. These rules, it should be noted, are never presented to the electorate for approval. Nevertheless the President himself has a very limited effect on government policy. This can be seen by the general continuity of policy, foreign and domestic, regardless of who occupies the office. It is unfortunate that so many Americans seem to see the President as a kind of King, who determines major changes and bears sole responsibility for the state of things during his tenure. It's a kind of mythology that obscures the true workings of the system, to which the President is more or less insignificant.
As far as social policy goes, the major lines are determined like any other type of policy, by the organized actions of interested groups. If there are enough people with enough commitment and resources working for a certain cause, and working effectively, pressure will be exerted on the architects of policy. If this pressure, which may include public opinion but might not, is sufficient it's likely to result in the kind of thing they want. All domestic policy works this way, whether the cause is civil rights or the abolition of the capital gains tax. Public opinion is basically irrelevant unless it is organized to exert pressure. Very small minorities can bring effective pressure to besr if they are organized, committed, and command the resources. The President will respond to this pressure just like any other part of the system. If he or she is too resistant, they will lose the support of thier party and that's it for them.
The only part of the government, in he US at least, that can hold out against this organization pressure is the judiciary. It has by far the largest impact on social reality of any govt branch, through long-lasting and unaccountable decisions which remain in force for hundreds of years, perhaps, unless overturned by another judicial ruling.