May states be required to grant exemptions for business owners whose Sabbath requires them to close their business on another day?
No. In Braunfeld v. Brown (1961) the Supreme Court held that observance of a Sabbath was an individual’s choice, and that a person was not discriminated against or disadvantaged by the state for its decision to require the closing of businesses on a day other than that individual's Sabbath. States may choose to allow exemptions for certain individuals, but they may not be required to do so.
May states choose only certain types of businesses to be closed on Sundays?
Yes. Where the state determines that a day of rest would be desirable in some kinds of businesses and not in others, they are permitted to restrict only those that they deem to be necessary. Likewise, the state may decide to forbid or limit the sale of certain items (such as alcohol) on any given day, so long as the decision is justified by some secular purpose instead of a religious one. In a 1999 decision, Harris County, Texas v. CarMax Auto Superstores, Inc., the 5th U.S. Circuit Court of Appeals upheld a Texas law that forbade car dealerships from being open on consecutive Saturdays and Sundays. Effectively this forced the business owners to choose one day or the other as a day of rest for their employees, though it did not dictate any particular preference as to which one should be adopted. The court denied that the law unfairly discriminated against car dealers or established any sort of preference for religion as opposed to no religion.
Are state holidays constitutional when they are directly tied to some religious observance?
The Supreme Court has declined to address this issue, though the lower courts
strongly favor the constitutionality of such holidays. The 9th Circuit in 1991
upheld legislation making Good Friday a state holiday in Cammack v.
Waihee, reasoning that the absence of a major traditional holiday in the
spring created a state interest in decreeing one, and that it made sense for the
legislature to select a day that would already be used by the majority of
citizens as a holiday. This decision set the stage for the 4th and 6th Circuits
to issue similar rulings. The 7th Circuit disagreed in Metzl v. Leininger
(1994), holding that because Good Friday is an exclusively Christian holiday
that has in no way been secularized, as have Christmas and Easter, its elevation
to the status of a state holiday was unconstitutional because it sent a message
of endorsement to the public, even if the practical result was neither to
advance nor inhibit religion. The holding in Metzl did allow for a
finding of constitutionality, however, if the legislature would merely make the
effort to advance a secular reasoning for the case.