1. Identify the compound and reaction

Calomel is mercurous chloride: $Hg_2Cl_2$

It reacts with ammonium hydroxide $\left(NH_4OH\right)$, which is essentially aqueous ammonia.

2. Known reaction of calomel with ammonia

Mercurous salts undergo a characteristic reaction with ammonia in which blackening occurs due to formation of finely divided mercury and an amido-mercury compound.

The reaction is: $Hg_2Cl_2 + 2NH_4OH \rightarrow Hg + HgNH_2Cl + NH_4Cl + 2H_2O$

So the important mercury-containing product formed is: $HgNH_2Cl$

This compound is called mercuric amido chloride.

3. Check the options

• A: $HgO$
  Incorrect. Mercuric oxide is not the product of this reaction.

• B: $Hg_2O$
  Incorrect. Mercurous oxide is also not formed here.

• C: $NH_2-Hg-Hg-Cl$
  Incorrect. This is not the standard product written for reaction of calomel with ammonium hydroxide.

• D: $HgNH_2Cl$
  Correct. This is the amido chloride formed in the reaction.

4. Final answer

Therefore, calomel on reaction with ammonium hydroxide gives: $\boxed{HgNH_2Cl}$

So the correct option is D.