Yes, once a proposed amendment has been ratified by the requisite number of states (3/4), it becomes the law of the land for the United States, as per Article Five of the Constitution. Of course, sometimes states or regions will violate the Constitution (especially after the Supreme Court interprets the status quo as being in violation of the Constitution, such as with Brown v. The Board of Education with found segregation to be illegal, but which did not take full effect in some places until the 1970s), but this did not happen with the 19th Amendment.

The amendment was challenged in Leser v. Garnett (1922), in which a man (Leser) sued to stop women from voting in Maine, which had not ratified the amendment. The Supreme Court found that the Amendment applied to state elections as well (although the 19th Amendment did not state so explicitly, only saying that "The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of sex"), just as the 14th Amendment had applied to state elections.

States that ratified the 19th Amendment after Tennessee did so symbolically--the law was already in effect. This is normal for Amendments to the Constitution. For Example, Mississippi, which did not ratify the 19th Amendment until 1984, also did not ratify the 13th Amendment (the amendment that prohibited slavery) until 1995. Nevertheless, the 13th Amendment, ratified by the requisite number of states in 1865, was in full effect in Mississippi well before 1995.

If you want to see how a ratified amendment can indeed fail, look at the 14th 15th Amendment, which guaranteed blacks the right to vote--but which was shirked by Southern states through poll taxes, literacy tests, etc., until Congress passed the Voting Rights Act of 1965, which authorized the Federal Government to go to the South and enforce the 15th Amendment by ensuring that blacks were given the opportunity to vote.