No, unfortunately that right was made invalid by the 1964 civil rights act. You or I can't demand that a baker sell us a cake and then have the baker heavily fined or put out of business if he doesn't, but a pair of homosexual perverts can. If that's not a specially created "right" for one type of people I don't know what would be.
The issue back at the time the Civil Rights Act was passed was primarily restaurants, hotels, and other businesses that refused service to black people. Congress decided and the courts agreed that the right of the state to regulate businesses trumped the right of the business's owner/operator in that particular situation, and I agree. I think that law was justifiable in those days because racial characteristics such as skin color and/or other physical features are usually determined by race and are irreversible even if the person in question wanted to change. But OTOH homosexuality has been proved beyond doubt to be reversible by literally tens of thousands of former practicing homosexuals who have come out of that perverse lifestyle and now live well adjusted normal lives with opposite-sex pardners and children. IMHO forcing people to go against their traditional American Christian beliefs simply in order to cater to the whims of that one particular portion of the population is wrong-headed and un-American no matter what the courts of today say.
I think businesses should have the right to serve or not serve anyone they please. To deny that right, as now-Senator Ran Paul once pointed out, is a denial of our property rights. Once big government gets its big foot in that door, all bets are off. The gay Oregon law is proof of that.