Not exactly. The first part is correct but SCOTUS has misapplied the 14th amendment to the bill of rights and now binds the states to the restrictions that were originally applied only to the federal government.
The 14th Amendment's due process clause, which states that no state can deprive a person of life, liberty, or property without due process of law, has been interpreted by the Supreme Court to include the liberties guaranteed by the First Amendment. This doctrine, called incorporation, makes states subject to the First Amendment, including its Establishment Clause, which prohibits the government from establishing or sponsoring religion. The Supreme Court explicitly applied the Establishment Clause to the states in the 1947 case Everson v. Board of Education.
Did Congress or the ratifying States intend to incorporate, in whole or in part, the Bill of Rights
tenthamendmentcenter.com