How can the Axiom computer know Earth is safe based on the one little plant EVE found?

The planet was toxic due to the waste that humanity created. As such, any soil and water would kill the plants that grew there, and it was no longer feasible for humanity to stay.

The plant is proof that Earth is ready to produce life again. The soil and the water are safe enough for a plant to grow with them.

It's not infeasible to think that the ship had a seed vault in it, given its phenomenal size, and original mission was to return to Earth at some point.

How can they know Earth is safe based on one little plant? Can one plant represent that an environment is healthy and safe?

Out-of-universe, it's a simple metaphor for the audience, and it provides a MacGuffin for the characters to chase after. Don't look too hard at the science of WALL*E.

In-universe, and in reality, it can't, and it doesn't. @BCdotWEB was correct in their comment, "It's not about Earth being safe, it's about Earth being salvageable." "Green plant == safe" is the simple plan sold to people who are used to having everything done for them. The reality is people are going to have to put in work.

As revealed in the film, the whole "return to Earth" plan was a sham. They don't expect Earth to ever support life again. But they needed to have some sort of plan in case anyone looked, just like they needed to seem like they're looking for life, to give people hope. It's an overly simplistic plan to be understood by people who grew up in a space mall; they're so out of touch with reality they think you can farm pizzas.

It's also clear that while B&L can think long term about technology, they're able to design a fully automated spaceship to sustain a large human population for generations, they're very short-sighted about the environment. They couldn't fix the Earth in a few years, so they don't expect it to ever be habitable again. They're not too worried about what happens if a plant is ever found, that's for future generations to worry about. If they do, the autopilot will cover it up. Problem solved forever!

The plant started as an indication that Earth was safe to return. Once it became clear that plan was a sham, it became was the key to get the Axiom to return home.

When the Captain sees Earth, he's initially incredulous. He expected everything to be taken care of for him, like on the ship.

Captain: Wait...that doesn't look like Earth. Where's the blue sky? Where's the grass?

Then he realizes they might have to put in some work.

Captain: There you go little guy. You came a long way for a drink of water... Just needed someone to look after you, that's all -- We have to go back.

The secret message from B&L makes him realize they just gave up on Earth.

B&L CEO: So just, uh...just stay the course. Rather than try to fix this problem, it'll just be easier if everyone remains in space.

Captain: Easier?

The Captain expresses his desire to do more than just live on a ship, he wants to go take care of the Earth. The plant doesn't indicate the Earth is safe, everything about the Captain's life has been safe. The plant rekindles his connection to the Earth and desire to do something significant rather than be entombed in a space mall.

Captain: Out there is our home! Home, Auto! And it's in trouble! I can't just sit here and...and...do nothing! That's all I've done! That's all anyone on this blasted ship has ever done...NOTHING!!

Autopilot: On the Axiom you will survive.

Captain: I DON'T WANT TO SURVIVE! I WANT TO LIVE!

The end credits show, not a return to a garden world, but working back up through the various stages of human technology.

Basically, the Captain got lucky. Green plants can and do grow in environments toxic for humans, short and long term toxic. It's also lucky he has an inexhaustible supply of robots to til the marginal soil with them.