D

$\because$在$\triangle ABC$中，$\angle BAC=90^{\circ}$，$AD$是斜边上的高，
$\therefore \angle ADC=\angle ADB=90^{\circ}=\angle ADE+\angle BDE$，
$\therefore \angle B+\angle C=90^{\circ}$，$\angle C+\angle DAC=90^{\circ}$，$\angle B+\angle BAD=90^{\circ}$，
$\therefore \angle B$，$\angle DAC$与$\angle C$互余，$\angle BAD$与$\angle B$互余.
$\because DE\bot AB$，
$\therefore \angle BED=\angle BAC=90^{\circ}$，
$\therefore AC$∥$DE$，
$\therefore \angle C=\angle BDE$，
$\therefore \angle C+\angle ADE=90^{\circ}$，
$\therefore \angle C$与$\angle ADE$互余.
综上所述，不与$\angle C$互余的是$\angle BAD$.
故选：$D$.