Q: Is there any way to get iOS 6 for my iPhone 4?

I have no security software on this computer.

I'll try restoring on another computer.

So far so good.

I'm afraid this phone is dead.

I have not tried putting the phone in recovery mode manually. Since it already does go into recovery mode, and I am able to start restoring, why do you think putting it into recovery mode manually would yield different results?

I have now seen that recovery mode screen on my phone several times. I'll try the manual method now.

After putting the phone into recovery mode manually, the steps I'm observing are the same steps I've already seen several times.

Just like before.

Well look at this! The trouble I reported above started after I was out in the rain last night using the Maps app on my phone to find my way around lower Manhattan. Perhaps the rain was too much for the phone and if I had waited longer it might have dried out without me needing to install iOS 7, but it's too late now.