I am running a campaign during which all the gods left for various reasons. Because of this, I changed up the Paladin and Cleric classes, by providing an alternate source of power for paladins which changes some of their mechanics, and by providing an alternative class in the form of alchemists for clerics.

At this point, nobody had a cleric or paladin player, and none of the players had shown interest in playing one up to this point, so I thought I would have no trouble.

Most of my players are fine with this and are looking forward to the new setting, but one of my players says that Clerics do not need gods to cast spells and won't quit arguing about it with me. I disagree, and I have tried reasoning with him, but nothing is working. Any suggestions on how to get them to stop arguing with me, or any other solutions?

“That is the premise of the campaign I’m running. If you don’t think that you can enjoy this campaign, you are welcome to join us for the next one.”

The spells of clerics, druids, paladins and rangers are called divine magic. The spell casters' access to the Weave is mediated by a divine power - gods, the divine forces of nature, or the sacred weight of a paladin's oath. ("The Weave of Magic", PHB p. 205)

Not all divine powers need to be derived from deities. ("Forces and Philosophies", DMG p. 13)

As discussed here and here among other places, Gods not existing does not necessarily mean Clerics cant cast spells. So your player could be right going by the standard rules.

However, you as the DM have final say about the world, classes, spells etc. If Clerics simply don't exist in your world just plainly explain this to your player. Just as some races are not available in some campaign settings, Clerics and Paladins are may not be available in yours.
Before making major alterations to the standard rules though (such as completely removing one or more class options) you should consider if this is entirely necessary. How committed are you to your 'no clerics' rule? How much does your player want to play a Cleric? Would your player settle for another magical class? Is a compromise possible, such as providing a source for Cleric magic that isn't a god ?

You should also make sure any restrictions of your setting are clear to your players before you start your campaign.

You could stay firm to your ruling, and are in your right as a DM to do so, but consider, if you have a setting that is 100% to your liking but have no players to enjoy it with, what good is the setting?

You kick them out or you work with them

If you want to run a campaign that has no divine magic, that is fine, you're in your right to do that as a DM and if the player doesn't want to play in that campaign, they can leave. However, in one of the comments you mention that this was an already going campaign, in which case, changing core concepts halfway through is a pretty bad idea©.

Imagine you are playing a Wizard, and you've been building up to level 6 the entire time in your Necromancy school, expecting some kickass skeletal guards to follow you around. Just when you hit level 6, your DM tells you "Yeah in my campaign, the dead can't come back to life, so you can't do that."

That's perfectly valid from a "the DM decides the setting" perspective, but it's a huge blow to the player's character concept, and you've essentially just crippled that player's fun because you are making fundemental changes to the rules everybody plays by, without informing people at the time of character creation.

So my suggestion would be to simply work with the player, the rules suggest that you don't need gods for divine spellcasting to work. Divine spells still work in places that don't allow you to contact your gods such as Ravenloft, so you can easily work something out with your player to allow them to play a Cleric like they imagined, without having gods in your setting.

I have tried reasoning with him

That's your first mistake! You're the GM, if you say they do, they do.

However, as a side note, I would hope that you've compensated the Cleric class in some other way for their inability to cast spells. He may (justifiably) now feel that it's no fun playing his character, and if this is the case, it's definitely something you need to address.