C

$\because 3＜\sqrt{10}＜4$，
又$\because {|x-\sqrt{10}|}^{2}$取到最小值，$x$为整数，
$\therefore$整数$x$的取值要更接近$\sqrt{10}$，
$\because 3$更接近$\sqrt{10}$，
$\therefore$当${|x-\sqrt{10}|}^{2}$取到最小值，$x=3$.
故选：$C$.