Why are silver halides insoluble in nitric acid The silver halides $\ce{AgCl},~\ce{AgBr},~\ce{AgI}$ are insoluble in $\ce{HNO3}$. This is given in J.D.Lee's inorganic chemistry text. (The strength of nitric acid is not specified though).

But won't this reaction take place?
$$\ce{X- + HNO3 -&gt; X2}$$

This example is taken from JD Lee's book itself:
$$\ce{6Br- + 8HNO3 -&gt; 3Br2 + 2NO + 6NO3- + 4H2O}$$

because  concentrated Nitric acid is a powerful oxidizing agent.
Nitric acid (even concentrated) is essentially an aqeuous solution.

Effectively, silver halides will not dissolve in water (they  have incredibly low $K_\text{sp}$), so the reaction indicated is moot.... there is no appreciable concentration of bromide in solution to be oxidized.

$K_\text{sp}(\ce{AgBr}) = 5.0\times10^{-13} \Rightarrow [\ce{Br-}] \approx 700~\text{nM}$