This should fall under a reasonable right to privacy. They should need to get a warrant for a specific crime to search for specific plates, not blanket access to search all and contrive crimes stridently after the fact.
See it’s funny: while one can argue about “unreasonable”, perhaps, the very clear words “shall not be violated” seem pretty unqualified-but people don’t like he very clear words “shall not be infringed” either so :shrug:
Given that this should fall into the few relationships that have historically been treated as sacrosanct-a man and his priest, doctor, and counsel/attorney-it’ll be a hard uphill climb to make a case for investigation into/prevention of medical care decisions not being an unreasonable search or unreasonable seizure. Then again IANAL, so again who knows what fun reasoning lawyers will come up with :shrug-again:
4th: “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.”
Your opinion presupposes abortion is a medical procedure and not a murder and ever larger percentages of the population would disagree with that categorization the closer to term a pregnancy is.
You also don't seem to have a good grasp on how the constitution works. It has a mechanism for amending itself so it is most appropriate to interpret it in an originalist framework. That is usually interpreted as meaning what the people who wrote it likely meant or, if referring to something in common usage at the time, then the modern item currently in common usage or likely to be in common usage. Abortion was illegal at the time of writing the constitution and also at the time of writing the 4th ammendment and stayed illegal for many decades following so there is no way they meant that procedure was protected by the constitution, it was and is very clearly the purview of the state legislatures (or you can amend the constitution). This is why overturning roe was the right thing to do even if you strongly support the legality of abortion, as I do.