Unfortunately there's nothing within the rules that dictates how a Druid must act. Though, the introduction to the class clearly states

Druids are also concerned with the delicate ecological balance that sustains plant and animal life, and the need for civilized folk to live in harmony with nature, not in opposition to it. (PHB, pg.65)

Druids are also part of larger organisations, called Circles, the description of which includes this line:

Druids recognise each other as brothers and sisters. Like creatures of the wilderness, however, druids sometimes compete with or even prey on one another. (PHB, pg. 68. Emphasis mine)

There's no information on how to punish a druid for living in disharmony with his or her environment, but it seems clear that they believe in the natural order; the laws of nature. Even when it means the strong hunt and prey on the weak. A druid is certainly not forbidden to kill animals, especially if it's for their own survival. Hunting and preying is the natural way.

And they are certainly under no obligation to constantly care for all plants around them. But perhaps going out of their way to destroy them would cross the line, though.

A Character's behaviour is not governed by their class

One thing to keep in mind that a character's class is a mechanical term and used within the game to determine their capabilities. Within the world, their vocation may be quite different. Perhaps the Bard is simply a charismatic politician, swaying the populace with powerful speeches instead of moving song. Perhaps the Wizard is actually a hedge mage, carefully tending his garden in solitude and communing with natural spirits instead of actively researching in a library, locked in a tower.

Acting stealthy, picking locks, and thievery aren't specific to a Rogue. Rogues simply specialize in such tasks. Plenty of warriors enter a battle rage, Barbarians have simply learned how to channel that rage to make themselves stronger during battle.

Decide what it takes to be a Druid within your setting, and discuss this with the player

Unfortunately, it seems you've allowed the player to assume the character class without fully discussing your setting and/or their backstory, and hence their role within your world. How did they gain their powers? Are they part of a Druidic order (By the book, they are in a Circle)? Are there repercussions from this order when a Druid acts out of interest?

Maybe the character isn't a druid at all?

One thing I should make explicit is the option that perhaps the character isn't a druid in your setting at all. After you discuss Druidism within your setting with the player, discuss other flavour-based options that allow him to keep the class but still take his character in another direction. From an example in the comments maybe the PC is from a barbarian tribe that uses nature-based magic (but to survive in nature rather than protect it). There are many explanations, as many as you can imagine, as to why your Player's character would have these powers without actually being a druid within the setting; mechanically a Druid, but not referred to as such by the other characters who inhabit your world.

There are some options available if you decide he's a druid in both name and function

After you discuss with the player how druids work in your setting, explain to them why, if his behaviour continues as it is, he will face certain consequences. Importantly, though, don't punish the player for his current "misdeeds," especially if you haven't already spoken. If, after you've had a discussion, the player continues behaviour disruptive to the natural world, make a story out of it.

• Perhaps their Cirlce becomes determined to hunt him. Either to bring him to justice and remove his powers, or to hunt and kill him like one would a mad animal.

• Maybe the god(s) or spirits of the natural world disapprove of their actions and deem them unworthy of their druidity and remove their powers until they repents. I certainly know certain Fey may have a problem with their behaviour.

• Maybe their magics start to go wrong, and the animals and plants he calls to his aid turn on him instead of acting as allies?

• Perhaps an investigation by the local guardsmen has led them to believe the PC is responsible for a recent robbery?

By the book, you can take inspiration from other rules on how to act

My one suggestion would be to treat it as a Paladin who has broken his oath. A short blurb on this may be found on pg.86 of the PHB, and there is an Oathbreaker Paladin in the DMG. Suggest the character play a fallen druid (if they can exist in your setting) and find/create an such a class with them if they find this method interesting.

In the end, it is your Player's Character

Perhaps suggest the player take class levels in either Barbarian or Rogue, as they may better fit their current playstyle, but do not force them to do so. There's nothing in the rules dictating their behaviour, and it seems to me they were not aware of your expectations.

But do take the opportunity to inspire yourself and your campaign and present challenges for the player and their party. All actions have consequences.

This is a place for a Charisma (Animal Handling) check

Influencing a beast to be friendly is a charisma check...

A charisma check might arise when you try to influence or entertain others

"I really want to use my animal handling proficiency though." The Player's Handbook talks about a variant rule enabling Skills with Different Abilities.

Normally, your proficiency in a skill applies only to a specific kind of ability check. Proficiency in Athletics, for example, usually applies to Strength checks. In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the GM might ask for a check using an unusual combination of ability and skill, or you might ask your GM if you can apply a proficiency to a different check.

If your GM allows it perhaps you can make a Charisma (Animal Handling) check. It is certainly reasonable to include your proficiency for animal handling when interacting with an animal. Make sure to check with your GM if you are using this variant rule though.

If not...

You should use Persuasion regardless of whether the spell is active or not. Here are what the skills do according to the Basic Rules.

Animal Handling

When there is any question whether you can calm down a domesticated animal, keep a mount from getting spooked, or intuit an animal’s intentions ... call for a Wisdom (Animal Handling) check.

Persuasion

When you attempt to influence someone or a group of people with tact, social graces, or good nature, the GM might ask you to make a Charisma (Persuasion) check.

"Befriending of a beast" is not within the basic purview of Animal Handling so you should use Persuasion (it would likely be close to impossible to even attempt this without Speak with Animals or a similar spell; you should at least have disadvantage since you don't share a language without the spell).

Wisdom checks overall are...

effort[s] to read body language, understand someone’s feelings, notice things about the environment, or care for an injured person.

This has nothing to do with communication, which is why you cannot make a Wisdom (Animal Handling) check for this.

If your GM is allowing animal handling to be used for befriending beasts, then there is no reason to force a player to change skills with the new communication method. The intelligence of the creature didn't change, so the method for befriending would be the same; you would just get the added benefit of verbal communication (possibly advantage depending on the situation)