Opinions: Thanks to Reynolds for the quick links. O'Connor, in a separate concurrence, makes that "rational basis" argument you discussed, looking toward equal protection rather than due process.

Texas attempts to justify its law, and the effects of the law, by arguing that the statute satisfies rational basis review because it furthers the legitimate governmental
interest of the promotion of morality. In Bowers, we held that a state law criminalizing sodomy as applied to homosexual couples did not violate substantive due process. We rejected the argument that no rational basis existed to justify the law, pointing to the government's interest in promoting morality . . . This case raises a different issue than Bowers: whether, under the Equal Protection Clause, moral disapproval is a legitimate state interest to justify by itself a statute that bans homosexual sodomy, but not heterosexual sodomy.

Meanwhile, Kennedy, writing for the majority, points to due process via Griswold:

In Griswold the Court invalidated a state law prohibiting the use of drugs or devices of contraception and counseling or aiding and abetting the use of contraceptives. The Court described the protected interest as a right to privacy and placed emphasis on the marriage relation and the protected space of the marital bedroom.

Which reasoning goes a long way toward explaining the GOP line that they'd like to see sodomy laws repealed, but not by the courts. They were loath to see someone like Kennedy affirm Griswold so clearly.