Yes, but that particular clause refers to protection under law as administered by an individual state, not the federal government. Read the first half of that sentence:

No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States

Clearly the sentence, in its entirety, is referring to states making or enforcing laws in conflict with the U.S. Constitution, since federal law has previously been covered in other Amendments (4 and 5 in this particular case). While the first part of Section 1 discusses citizenship as it applies to both state and federal authorities, the second part of that section refers to state laws that are in conflict with rights granted by the federal government.