Technically, you can say those things.
"The dog is aesthetic." "The world is aestheitc." "That tree is aesthetic." There is nothing wrong with these sentences.
The correct usage of the word aesthitic lies in what you mean to say by it.
Like Colleen V said, the word aesthetic is not a synonym of the word "beautiful" which is the actual state of having or being considered to have beauty.
Instead it means "relating to, or concerned with beauty, emotions, and the senses" (Source).
Keeping this in mind, there's only two ways that this word has been seen to be used in my experience, when being used as a descriptor for a noun:
- The noun is something which is meant to inspire, or capable of inspiring thoughts related to beauty.
- The noun is a living being capable of comprehending ideas related to beauty.
Now, if we apply these two rules to the second example you mentioned,
The world is aesthetic.
You could mean:
The world is an aesthetic place.
Meaning the place itself or the things that the place consists of, are capable of inducing thoughts relating to beauty.
Or assuming that by 'the world' you mean 'all beings in the world', you could mean:
The beings of this world are aesthetic ones.
Meaning that they are capable of considering, or like to consider ideas related to beauty.
Now, depending on what your opinions are, and what you are trying to convey by using the word aesthetic, the other two sentences can also be correct. Nevertheless, it is important to be clear about what one is trying to say, so if these sentences were to be truly "correct", they would use other words to explain how the word aesthetic applies in those particular cases. Like the way demonstrated above when the second sentence was used as an example.