Why did he not request a sign?
Rashi: He did not want Hashem's name to be sanctified via him.
Radak: This was due to lack of Emunah. Even though he spoke as if he believes [so he does not need a sign], the Navi knew his heart 1 . Therefore, he said "Sal'u Gam Es Elokai" (13).
Kli Paz citing Alshich: I will not test him - He can choose the sign He wants to give!
Malbim: If he believed, he would have said 'I will not test, and I will not ask.' I will not test for myself, for He is already tested in my eyes, and I have no doubt. I will not ask for others, for all trust in Him. Rather, he said "I will not ask, and I will not test" - I will not ask for others, for I do not want at all that His name be sanctified and others will believed in Him. I will not ask even for myself, for even with all the signs, I will not believe His words.
Why did he say "Veis David", and not call him by his name?
Radak: This was to disgrace him, like "Shma Na Ben Achituv" (Shmuel I, 22:12) He mentioned David, and not [Achaz]'s father, because he had kingship due to David, and Hashem does miracles for him due to David. Similarly, Moshe said to Korach 1 "Shim'u Na Bnei Levi" (Bamidbar 16:8).
Kli Paz citing Alshich: You did not want a sign. Therefore, the sign is not for you, rather, for Beis David.
If Moshe addressed only Korach, why did he say "Bnei (plural) Levi"? And if he addressed also others, this is why he said "Bnei Levi", and not to belittle Korach! (PF) Also refer to Shmuel I, 20:30:3:1 and the note there.
