I would rephrase it further. This is about the balance of powers in the government. The argument isn't that we don't have this right, it's that it isn't a Constitutional right.

Our existing Constitutional rights are more or less straightforward - "No one can prevent you from peacefully speaking your mind," aside from exceptions like fraud and credible threats. The judicial branch, the court system, is responsible for stopping wrongdoers and overturning laws that violate those rights.

By contrast, the proposed right, "No one can prevent you from having a stable climate where you live," is completely unenforceable by the courts.

The scope is too different: it's unclear what actions and laws would be in violation of that right. Would you be infringing on your neighbor's right to a stable climate because you drove your car to work when you could have ridden a bike? Is your city infringing on your right to a stable climate if it uses incandescent light bulbs in government offices, or fails to mandate solar panels on every roof?

The point being there is no Constitutional right to a stable climate because there's not really a way to directly violate that right in a way that the courts can enforce. Instead, it needs to be a policy decision passed by legislation with specific rules and actions in mind. That's a power reserved for Congress and not the courts.

By that logic, they cannot guarantee the freedom of speech either though. They can try very hard, and do their best to make amends for when it's breached, but many people have been silenced illegally by the US government. They can try, but they cannot promise this fact.

I don't see why one couldn't apply the same to climate.