State law defining a live, healthy birth as something without remedy at law does not offend state constitutional guarantees of open courts and jury trials.
First Amendment right to petition incorporates right to the courts, but states can define the boundaries of a claim.
Statute survives rational basis review under the federal right to privacy, as legislature had an interest in medical malpractice reform.
Seventh Amendment not incorporated against the states.
Gender discrimination claim insufficiently developed.
http://media.ca1.uscourts.gov/pdf.opinions/17-1997P-01A.pdf
Since 2011. An occasional commonplacing of the U.S. Circuit Courts of Appeal.
Neither an authoritative statement of the law nor legal advice. Quick work.