But people are being prevented from passing laws against abortion, which should be within their rights as participants in democracy.
No, because they're limiting the rights of people who don't
have religious objections to the procedure. Having a religion doesn't give you the right to impose it on others.
Not unilaterally, but democracy allows you to apply your personal principles, i.e. your free speech and religion, to the pursuit of legislating a good society. What you are arguing would allow mob bosses to insist that religion shouldn't give the people the right to impose the commandment not to kill against others who use killing as a punishment for disobedience, disloyalty, disruption of business, etc. People have the right to legislate laws against homicide/manslaughter/suicide/self-mutilation/etc. based on moral reason and so why should abortion be exempt from the same democratic legislative discourse?
As long as no one is being directed by the state to terminate their pregnancy there's no reason to restrict the practice because some religious people object to it. Occasionally there are medical reasons for a woman to obtain an abortion — outlawing it could result in the death of the woman.
Abortion was legal prior to Roe v. Wade but a police report of rape was required. What Roe v. Wade did was to allow women to protect the identity of the father, which protected men from culpability/liability for abortion. Nowadays, with DNA testing there is no way a man could be framed as the father of the fetus so there is no reason not to hold men accountable for impregnating women against their will.
The bottom-line with diversity is that religion prompts people to view fetal tissue as not just tissue growing in a female body but as a nascent human life, and therefore worthy of honor and protection. So if Dems really respected religious diversity, they would seek a solution to the abortion issue that would satisfy all people/religions, and that would ultimately involve taking a stance against not only non-consensual sex (which they currently pursue fervently via metoo/etc.) but also against all sex that carries any risk of unwanted conception.
In short, all people should refrain from coitus except when pregnancy is desired in order to avert the possibility of conception. This is as achievable as gun-safety that prevents firearms from causing murders and fatal accidents. When someone misuses a firearm and causes injury or other harm, they're held accountable; so why shouldn't the misuse of coitus resulting in abortion be similarly prohibited and punished?
If people are expected to avoid hunting accidents with guns, why not also to avoid sexual accidents resulting in pregnancy? Hunting accidents violate the religious commandment not to kill and misuse of sex also violates religious principles, so why not allow religious people to advocate laws against both abuses as well as any others that their consciences lead them to pursue legislation against?
Separation prohibits the establishment of state religion, which means that democracy and secular government are supposed to be the filters that prevent one church from ruling over all others. In democracy, all churches/religions are supposed to engage in dialogue to achieve consensus laws that respect diverse sensibilities about moral right and wrong. The idea is that there may be diversity of beliefs, but through democracy we can arrive at consensus standards and laws that can be respected and honored by all.