While there is no defined "right to privacy" in the Constitution, it may not be necessary to explicitly amend the Constitution to provide one based on two existing amendments.

It's not hard to make the argument that the 4th Amendment covers personal privacy. It says,

"Amendment IV - Search and seizure. Ratified 12/15/1791.

The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."

In addition, there's the 9th Amendment which may be more to the point. The 9th Amendment says,

"Amendment IX - Construction of Constitution. Ratified 12/15/1791.

The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people."

This amendment says that we have rights beyond those explicitly listed in the Constitution, which would include the right to privacy. These arguments are nothing new, as shown in 1890 article from the Harvard Law Review. While it isn't explicitly listed in the Constitution, the right to privacy was a key argument in the (in)famous 1973 "Row v. Wade" decision about abortion rights.