Also from the press release:
"Citizenship is a privilege, and only those who respect our immigration laws should be allowed to enjoy its benefits," said Sen. Paul. "This legislation makes it necessary that everyone follow the rules, and goes through same process to become a U.S. citizen."
Vitter and Paul do not believe that the 14th Amendment confers birthright citizenship to the children of illegal aliens, either by its language or intent. This resolution makes clear that under the 14th Amendment a person born in the United States to illegal aliens does not automatically gain citizenship.
Of course, it takes a very high bar to amend the Constitution: Both houses of Congress must pass a proposed amendment with two-thirds majorities, and then the amendment must be ratified by three-quarters of the states (38 out of 50). The president plays no direct role in the amendment process, but can of course advocate for or against proposals, and use his political capital in other ways.
Birthright citizenship, a right outlined in the 14th Amendment after the Civil War, was intended at least in part to guarantee citizenship for freed slaves and otherwise secure legal equality:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
Without the concept of birthright citizenship, it is possible for someone to be born without having citizenship in any country at all.
Numerous Republicans, however, have argued that it should not be construed so as to apply to the children of illegal immigrants, and have proposed various solutions ranging from a full-fledged constitutional amendment to a lesser legislative workaround. In the case of the latter, experts have told TPM that any plans to change that right purely through legislation are "clearly unconstitutional."