  We determine the exact value of the $2$-color Ramsey number of a connected
$4$-clique matching $\mathscr{C}(nK_4)$ which is a set of connected graphs
containing $n$ disjoint $K_4$. That is, we show that $R_2(\mathscr{C}(nK_4)) =
13n-3$ for any positive integer $n \geq 3$. The result is an extension of the
result by (Roberts, 2017) which gave that result when $n\geq 18$. We also show
that the result still holds when $n=2$ provided that $R_2(2K_4) \leq 23$.
