Consider a finite $m\times m$ board where every intersection either has a stone, or does not. If there are at least $2m$ stones on this board, then I claim there will exist three stones which are arranged in an axis-aligned right triangle.
To prove this, number the stones from $1$ to $2m$. For each $1\le k \le 2m$, let $f(k)$ be the number of rows spanned by the stones numbered $1$ to $k$, plus the number of columns spanned by these stones. Note that $f(1)=2$, $f(2m)\le 2m$, and $f(k)$ is a weakly increasing function. It cannot be strictly increasing; if it were strictly increasing, you would have $f(2m)\ge 2m-1+f(1)=2m+1$, contradicting $f(2m)\le 2m$. This means there must be some number $k$ for which $f(k)=f(k-1)$. But this means that stone number $k$ is the same row as a previous stone, and in the same column as a previous stone, so that these three stones form a right triangle.
In particular, if an $m\times m$ board is filled with stones in $n$ colors, then some color will appear on at least $m^2/n$ of the stones, so if $m^2/n\ge 2m$, there will be a right triangle of stones in that color. Therefore, for all $n$, an infinite colored board will have a monochromatic coloring, and to find one, you only need to search in a $2n\times 2n$ sub-square of stones.
This problem seems to fall under the topic of Ramsey theory.
As a side note, you can also prove that an $m\times m$ board with only $2m-1$ stones contains an axis-aligned right triangle. This is tight, since you can place $2m-2$ stones without forming a triangle.