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.