I think that the Supreme Court will side with the baker, Jack Philips.  This case is similar to
the Hobby Lobby decision in June, 2014 which went against the Obama Administration which
was attempting to force Hobby Lobby to offer contraceptive services as part of the medical coverage
provided to their employees under Obama care. Hobby Lobby objected as offering contraceptive services was contrary to their religious beliefs against birth control.
The First Amendment  of the US Constitution  guarantees the free exercise of religion.
This means that if your religious beliefs prohibit you from supporting the gay life style, the government
cannot force you to violate your religious beliefs/prohibitions as this would interfere with the
free exercise of your religious beliefs and practices.  The Civil Rights Act of 1964 outlawed discrimination
based on race, religion, color, sex or national origin and ended segregation. The law says nothing about
discriminating  due to religious beliefs.