"Yes", but the question is ill-posed
Stripped to the bone, the question boils down to:
Country X has laws that allow prosecution under some circumstances. Country Y does not. Could a national from country X claim asylum in country Y?
To which the answer is: it matters little why or how the laws of the countries diverge. What matters is country Y’s laws about asylum.
For example, in france, asylum is defined by this section of the immigration and asylum code. Excluding the case of war refugees, asylum requires that the person:
- (L712-1) is exposed to death penalty, torture or inhumane treatment, or a grave, direct and individualized threat against them due to civil unrest;
- (L712-2) is not under serious suspicion of having committed war crimes or a serious non-political offense;
- (L713-3) cannot relocate within their home country to a place where they would be safe.
Your hypothetical case would face difficulties under these criteria:
- Mere fines or short periods of imprisonment do not suffice to establish a grave threat under L712-1. For instance, Dorota Rabczewska was condemned by a Polish court to a fine slightly above 1000€ for saying that "It is hard to believe in something [the Bible] written by people who drank too much wine and smoked weed". She is a successful entertainer for whom the fine is pocket change, and Polish police would defend her if Christian fundamentalist militias tried to kill her; she would most likely not have been granted asylum in France if she had gone there after her trial. (That the court was reversed by the ECtHR later is irrelevant to this analysis.)
- It is questionable whether hate speech laws are political offenses for the purpose of L712-2. They are not under ECtHR jurisprudence (see the links at the end of the Wikipedia article about article 10). You might disagree, and you might be right in a philosophical sense, but what matters is what the court of whichever place grants asylum decides.
A contrario, the Constitution of Russia is mostly aligned with those of "Western" secular democracies, and contains provisions for freedom of expression. Nonetheless, that piece of paper does little to protect political dissidents from extrajudicial killings, "accidents", or zealous prosecution of non-political (fabricated) crimes; many of them have successfully claimed asylum in other countries.
Are you mixing things up with extradition?
Imagine that John Doe is a national of Blasphemistan, a country with strict blasphemy laws. He makes public comments that clearly break those laws, and judging by previous similar cases, can expect at least ten years of jail if he is caught. He escapes the country through the land border to Atheistan, and is caught by border police there. Atheistan is a secular democracy with no blasphemy laws.
Blasphemistan requests that Atheistan hands John over to them for prosecution. Atheistan should refuse such a request, because of the double criminality requirement: a person currently under country A’s power will not be handed over to country B on the basis of a crime that does not exist in country A. In that sense, the First Amendment is a bar to extradition.
However, it does not follow that A has to grant John asylum. I suspect that if John is a dual national and has a passport from Neutralistan (another secular democracy where John would be safe from prosecution from B), A could deport him to Neutralistan. But that is another question.
