Nothing in the rules state any sort of verbal component.
True, I guess they could silently call upon their god. So less unambiguous proof for an observer?
The effect just happens.
Plus we know it’s not a spell because it cannot be counterspelled and even functions in dead magic zones.
I guess we can infer it is not a spell because it is the effect of a spell. I don't see any specifics on whether such spell effects can happen in dead magic zones or not so that seems a DM ruling issue.
This whole meme about how gods are unknowable in DnD is so bizarre. You literally have gods appearing. They objectively exist. No believing in a god in a DnD world (typical world mind you, there are exceptions) is flat earth territory.
Ok. Since literally showing up and presenting yourself isn’t proof, what would count as sufficient?
Proof is hard.
There is little about D&D gods that is unique to a god as a god.
Magic is not proof of a god. It is proof of magic in a high magic cosmology.
Magical power of a god is not proof of a divine status. It is just proof of magical power in a context where lots of beings have magical power.
Granting magical power is not proof of divine status. Warlock patrons grant magical power and are generally considered a different class of powerful supernatural beings than gods (though this can vary from campaign to campaign and cosmology to cosmology such as the different treatments of Asmodeus as a god or not and a fiend patron to warlocks).
In part it will depend on how you define gods and what the natures of the gods are.
This is going to vary from campaign to campaign under the core rules.
Under the core rules a god might never show up. Even clerical divine intervention when the objectively real god is intervening might just be a subtle magical effect that the cleric might not be able to notice.
Under the core rules gods might be manifest in the world, live in a specific place where you can go see them, and show up for divine intervention in the flesh in showy ways.
Gods might be transcendent and conceptual beings. They might be concrete powerful magical beings of a specific sort. It can vary.
So someone believing that gods definitionally are immortal and transcendent could look at the FR gods and see them as powerful beings who call themselves gods but who can die and are not actually really gods.
If the proof you have about gods is clerical magic and stories of the gods there is room to be skeptical about the existence of gods. Magic such as true resurrection happens in 5e without gods (True Resurrection is on the bard spell list) and so a tradition of spellcasters who claims their magic comes from gods is just a group with claims. Their magic is real, and they assert their claims, but it is just a story which might be true or not. In 3e FR the Red Wizards of Thay said they were not bad guys, they were just out to sell magic items in foreign nations. A story to be believed or not.