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No. It says that "the said Constitution shall be never construed to authorize Congress to (...) prevent the people of the United States (...) from keeping their own arms.". The BoR was meant to limit the power of the federal government, period. When in 1824 the SCOTUS was asked whether the Fifth Amendment denied the states the right to take private property for public use without justly compensating the property's owner it unanimously announced without even hearing the arguments of the City of Baltimore that the limitations on government articulated in the Fifth Amendment were specifically intended to limit the powers of the national government. As Chief Justice Marshall said:

"The powers they conferred on this government were to be exercised by itself; and the limitations on power, if expressed in general terms, are naturally, and, we think, necessarily, applicable to the government created by the instrument. They are limitations of power granted in the instrument itself; not of distinct governments, framed by different persons and for different purposes.(...)

The third clause, for example, declares, that 'no bill of attainder or ex post facto law shall be passed.' No language can be more general; yet the demonstration is complete, that it applies solely to the government of the United States. (...)

Had congress engaged in the extraordinary occupation of improving the constitutions of the several states, by affording the people additional protection from the exercise of power by their own governments, in matters which concerned themselves alone, they would have declared this purpose in plain and intelligible language." There is no reason to suspect that the Second Amendment is different.