Yes, you are not allowed to try to foist off your morality on others. That is why Christians do not have a patent on marriage. Their claims to do so are based upon an event that demonstrably did not happen and worse yet they do not even apply it as their holy book does.
We have to be careful here, as some moral rules are indeed enforced, and generally agreed with, like murder. Other moral rules are associated with a religious belief that is not generally shared, like not being allowed to open shops on Sunday. In any given case, where does the rule fall?
The two examples I gave are clear cut, though I suppose you could argue that many rules based on morality would be applied in a totally secular society and religion didn't invent or own them. (To our gay friends, I totally support your right to be whatever you are and to act accordingly. Just so you understand!)
On the other hand, it's very misleading to brand something like abortion a totally religious/secular thing. I suspect that in a secular society there would be a section of the population that would oppose abortion and try to get it banned or controlled. Hopefully they wouldn't argue that single cells were people, but whatever. I do understand it, though I'm on balance pro-choice. For example, I would support much stricter laws against the ill treatment of animals, and would enforce vegetarianism if I had the power. I feel I have the right to work in that direction if I choose.