Of course there are cultural differences as well, but modern-day Spartas with 90% permanent underclasses are fundamentally repugnant under any culture.
I’ll also note that in many of the wealthy Gulf states, the rules are so unequally enforced that the ruling people do enjoy many of the vices they supposedly disagree with in Western culture. There are other countries where cultural differences are in good faith, Qatar isn’t one of them.
So, yes, in a sense it's just a matter of culture. The universe doesn't care either way.
But for entertainment value, to make a different argument against moral relativism: A higher proportion of people from Qatar would like to e.g. live in Europe than the proportion of Europeans who would like to live in Qatar, and vastly higher if we take relative social standing into account.
I'm sure there are plenty of ethical questions where a hypothetical, neutral outside observer would have trouble choosing. There are more ethical question where the neutral outside observer would easily be able to discriminate what cultural traits they preferred, while at the same time understanding why those sub-optimal cultural traits came to be and play a role in that particular society. Imagine sexual mores in a society with no birth control, lots of hard labor and the preference for peaceful succession due to early accumulation of capital (farms), for instance. It's easy to understand what role traditional and restrictive morals has in such a society, while at the same time acknowledging that living within the morals of such society would be a giant step down in quality of life.
So that's the complex argument. But that all ethics is generally speaking relative from a human perspective, can be trivially disproven by pointing out ethical standards that almost all societies except those that have them will find repugnant. Consider e.g. genital mutilation or institutionalized child abuse.
The fact that some number of people agree on some ethical standards, even almost everyone, does not do any work to prove their accuracy outside human opinion and experience, much in the same way almost all societies except those who believed in the Abrahamic God found others who did not to be (repugnant/heathens/blasthemers/barbarians/baby killers).
As a concrete argument specifically dealing with Qatar, there is a very long list of pragmatic reasons why the death penalty for adultery is Bad, assuming reasonable terminal values like "happiness is good".
You are also free to pursue an academic career, not take drugs, and speak elegantly. But that lifestyle is far less glorified in Western media than drug use, gangsta life, and profanity.
Here's a thought: his many Western popular media heros use violence to achieve their goals? How many use rhetoric, critical thinking, and respectful verbal persuasion?
I think you'll have a hard time arguing that the country with the death penalty for adultery is a less violent culture than the one that made Harry Potter and Doctor Who (both nonviolent).