Here's my take on the problem.
Say there's are a pair of cultures, each celebrating a celestial event like Equinox or Solstice. In culture A, the tradition is that at solstice you should wear red shirts. Suppose they're not extreme about it, but they may look at you concerned or disappointed that you're not respecting their tradition for wearing a red shirt on that day. In culture B, the tradition is to wear a blue shirt instead.
If you go to culture B (from culture A), and wear a red shirt at the day, people will find it strange and be disappointed.
Does that mean one of those cultures is right in an absolute sense, and the other is wrong? I think at least in a "good enough" sense, maybe not. I think in many cases it comes down to non-unique solutions -- more than one culture can be good. Either case may be fine.
However, I think we can with progress in philosophy, science, etc.. critique cultures as a whole: we can try to define what is good. Not only that, but I think in the sense of what is good, then yes: I think we should converge on common, more or less universal principles. Within the same principles, a universe of cultures can exist; furthermore, variety itself is arguably part of goodness. If everything is the same, then we might be missing out in serious ways, both in immediately practical terms of different viewpoints fostering new innovations (if everyone thinks exactly the same... everyone has mostly the same ideas), and also in a cultural, aesthetic sense of life being more interesting with more than one 'cultural way', so to speak. Yet cultures should mostly agree on common universal principles (at least in some 'assymptotic', aspirational sense -- principles should be agreed more and more, and disagreements ought to become more minor).
You may observe 'Ok, but those principles are just the product of the culture you started with!'. I think that's somewhat fair. But I think universal principles are kind of 'natural' in a certain sense -- I think there is a unique set of principles that is capable of unifying concepts like science, truth, morality, sustainability, art, culture -- as far as reason can go. If you look at mathematics, it is universally agreed whether a statement should be true or not. And we don't expect a different culture to find it differently. It's really surprising, I think, that reason should be able to tell anything about culture, but, although it's in some senses extremely difficult (specially if you're thinking of proving anything practical/concrete, like whether an art piece is beautiful or not), those things should all be accessible to reason. This is in no small part because the processes that happen in our own minds can be studied -- and the study of mental processes is in the end what can provide a great unification of all sciences and arts (although in practice like I said it's too difficult in the near term). We can understand the nature of happiness, unhappiness, beauty, motivation, and so on, in a very solid way, in the same way we can study the behaviors of particles, control systems, processors, materials, knots, and so on (there are some important details to this study that are too long for this discussion[1]).
(I think both Chomsky and Focault would probably agree on this point eventually)
That's not to say you can't have cultures that refuse all that, potentially until they 'die'. It's also not to say you couldn't have some weird culture that does mathematics in a different way and they try to persistently live with grave inconsistencies. This would make their mathematics fragile, and probably far less useful -- the point where we call it 'totally unreasonable, pointless'. But it doesn't seem impossible that they could be totally unreasonable. This seems to undermine the notion of universality. If universality doesn't mean all cultures share the values, not even that all cultures would converge or even agree on those values, what could it mean? I think universality only means (in the context of ethics) that a culture with a sufficient set of tools/principles, procedures and systems do eventually converge on common principles. It's basically a basin of attraction of principles (and practices), that needs a minimal set for bootstrapping.
So what is this minimal set? And critically, do we even have it?
I think it can be summarized as consistency, or truth. By requiring that language be logically consistent, and enforcing the notion of truth (across reason[2], individual and collective decision-making), plus of course being sentient beings ourselves (that allow us to recognize and understand truths related to sentience, which I think are at the root of all that matters[3]), we get science, we become able to talk with each other reasonably, we become able to agree, we become able to investigate, etc.. By demanding that ethical principles have some (and eventually rigorous) consistency, we can get to moral truths: if my (literal) brother and my neighbors are both human beings, why should I only help my brother and not my neighbor? How am I fundamentally different from others that I should be more important? (egotism), etc.. What makes it hard, fundamentally, is that sometimes those demands come into conflict with our instincts, or even major aspects of our society, so we can dangerously veer off this path.
I think we have it, and another name for it might be Enlightenment, both in the sense of western culture, and eastern cultures (e.g. in buddhism). Are we enlightened yet? :)
(I would say we're mostly on the path to enlightenment, although we haven't yet been able to discover and agree on some principles and truths that are very important to our lives -- we are on the path but still far from the ideals)
This is such a beautiful and important notion to me, I think it might be worth it to call this God (i.e. this enlightenement, this path to truth/science/everything that is good that can be known) -- with capital G, and why not worship and protect this notion.
[1] The difference is that you need sentient beings and our experiences of ourselves to be able to understand truths about sentience. In a way we are, and have to be, both subject and scientist, and this is a little different from strict views of science.
[2] I've talked so much about reason/truth that I think I should mention intuition too. Intuition is the practical means of thought. Intuition can be thought as large leaps of logic that may be uncertain and/or non-trivial. Most of our language statements don't actually follow from one another immediately in the sense of logic, instead they tend to jump many steps at once, or we even make propositions that are unproven (in mathematics we tend to use intuitive leaps to find proofs and then fill in the gaps with formal statements). Many things in life are actually not so important that we need to prove an ideal solution, usually good enough is enough. Still, this process and language itself relies on the infrastructure of logic, truth and consistency to work; and in many cases, involving important decisions, we can reduce uncertainty and improve decisions by refining our steps, and giving more solid arguments (approaching truth with greater confidence).
[3] Just to be clear, I'm far from proposing we become naive logical machines. Those would probably be both ineffective, and neglect taking into account the experience of being itself. Even fantasy is important in this sense, but I think we should be careful to keep fantasy separate from reality as much as reasonable (phenomena like 'Santa Claus' and psychological quirks notwithstanding). Fantasy is good and reasonable in a way! Same goes for things like arts, games, sports, leisure. Should be made from self-sustainable good feelings.