RALEIGH, N. C.—Republican Gov. Pat McCrory said Tuesday that he will seek to roll back some provisions of a new state law that limits antidiscrimination protections for lesbians, gays and transgender people in an attempt to “affirm and improve” the state’s commitment to equality after weeks of protest, including some corporations dropping plans to invest in North Carolina.

But Mr. McCrory said he was standing by a “common-sense” provision of the law requiring transgender people to use the public bathroom corresponding to the...