I am sure there is an obvious answer, but suppose I have a monic polynomial with non-zero, integer coefficients given by
$$P(x) = x^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$
Veita's Formulas state the roots are related to the constant term $a_0$ where
$$a_0 = (-1)^nr_1r_2\dots r_n$$
where repetition is allowed. Now suppose there exists a rational root $r_1$. Since the Rational Root Theorem states that all rational roots of a monic polynomial are integers, the root $r_1 \in \mathbb{Z}$. Does it follow that if $GCD(r_1, r_2r_3\dots r_n) = 1$, then for any other rational root, say WLOG $r_2 \in \mathbb{Z}$, that $GCD(r_1, r_2) = 1$? I am assuming it does, but I am not fully sure.
Your guess is right: for all $s,p\in\Bbb Z$ such that the roots of $x^2-sx+p$ are roots of $P(x)$ (in particular for $s=r_1+r_2$ and $p=r_1r_2$ where $r_1,r_2\in\Bbb Z$), $$\frac{P(x)}{x^2-sx+p}\in\Bbb Z[x],$$ in particular $$t:=r_3\dots r_n=\pm\frac{a_0}p\in\Bbb Z.$$ Therefore, if $\gcd(r_1,r_2t)=1$ then $\gcd(r_1,r_2)=1.$