$\begin{array}{l}\text{Here, it can also be written as}\\ \sqrt{{\text{(6)}}^{\text{2}}\text{+ (}\sqrt{\text{7}}{\text{)}}^{\text{2}}-\text{2.6}\sqrt{\text{7}}}\\ -\frac{2}{\sqrt{\text{(}\sqrt{\text{7}}{\text{)}}^{\text{2}}{\text{+ (3)}}^{\text{2}}\text{+2}\sqrt{\text{7.3}}}}\\ \\ \text{Apply the formula}\\ [{\text{ a}}^{\text{2}}{\text{+ b}}^{\text{2}}-{\text{2ab =(a- b)}}^{\text{2}}]\\ \text{and }\left[{\text{a}}^{\text{2}}{\text{+ b}}^{\text{2}}{\text{+ 2ab = (a + b)}}^{\text{2}}\right]\\ \\ \text{so, }\sqrt{\text{(6}-\sqrt{\text{7}}{\text{)}}^{\text{2}}}-\frac{2}{\sqrt{\text{(3+}\sqrt{\text{7}}{\text{)}}^{\text{2}}}}\\ \\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2}{3+\sqrt{7}}\\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2}{3+\sqrt{7}}\frac{(3-\sqrt{7})}{(3-\sqrt{7})}\\ \\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2(3-\sqrt{7})}{(9-7)}\\ \\ \text{[}\because {\text{a}}^{\text{2}}-{\text{b}}^{\text{2}}\text{=(a+b)(a}-\text{b)]}\\ \\ \text{=(6}-\sqrt{\text{7}}\text{)}-\text{(3}-\sqrt{\text{7}}\text{)}\\ \\ \text{= 6}-\sqrt{\text{7}}-\text{3 +}\sqrt{\text{7}}\text{=3}\end{array}$

Hence, option D is correct.