WASHINGTON — Battles over health care and religious rights are sure to continue, even after the Supreme Court ruled Monday that family-owned for-profit corporations like Hobby Lobby are not required to provide insurance coverage of contraceptives to their employees if the companies object on religious grounds.

About 50 cases involving nonprofit organizations and a similar number involving for-profit companies are pending in federal courts around the country, and many of those plaintiffs intend to push forward with the argument that they should be able to opt out of providing or authorizing coverage that conflicts with their religious beliefs.

Perhaps the most prominent of these cases, a challenge to the contraceptive-coverage requirement involving an order of Roman Catholic nuns, the Little Sisters of the Poor, could hinge on the reasoning cited by Justice Samuel A. Alito Jr. in his majority opinion in the Hobby Lobby case.

In his opinion, Justice Alito said the government had other, less restrictive ways of achieving its goal of ensuring access to contraceptive coverage. He pointed, for example, to an accommodation devised by the White House for certain nonprofit religious organizations, like hospitals and universities, that have “religious objections” to providing contraceptive coverage.