Your proof, as it stands, is not correct. Some particular issues:
You get to a point where $x = 3y + 3/2$ and $y = x/3 - 1/2$ and state that "since both $x,y$ are not integers..."; this assumes the conclusion that you're aiming for.
It's not true that both $x$ and $y$ are not integers. You could have one of the two be an integer, and the other not be an integer. For example, $(0, -1/2)$ and $(3/2, 0)$ are both solutions with $x$ or $y$ integral, but not both.
Its not convincing that $3y + 3/2$ is not an integer (and the point immediately above shows that that claim is not true!).
Stylistically, your assumptions are not explicitly stated and there is no introduction to the proof. You haven't stated that you're assuming $(x, y)$ to be a pair of integers solving a particular equation. Therefore, it's not clear how the contradiction is reached; at a minimum, you need to form the negation of the statement and clearly include assumptions.
So unfortunately, this is not a properly written proof. But it can be fixed without too much work; here's an outline to follow:
1) Introduce the players. Say "We proceed by contradiction. Assume that $x, y$ are integers such that $2x - 6y = 3.$
2) Isolate one of the variables and get the contradiction. Perhaps "Then $x = 3y + \frac 3 2$. Since $3y$ is an integer (why?), the sum $3y + \frac 3 2$ is not an integer."
3) Tell the reader why this is a problem. "This contradicts the assumption that $x$ is an integer."