you can put tape on it if it is the only nick it wont hurt
use a good electrical tape

Would nicking the only white wire cause the breaker to trip if white was truly being used as the neutral?

If the conductor (copper part) is damaged, it should be replaced. While the neutral is at "ground potential" with respect to voltage, the neutral does carry full circuit AMPERAGE and is as subject to heating as is the "hot" wire. A narrowed area due to damage can heat at that point. This condition will not trip a circuit breaker. But the heat can be intense enough to start a fire. Yes, I would tape a neutral if only the insulation is damaged. If the copper is damaged, replace.

If the breaker tripped you hit the hot(black) wire and either the ground(bare) or the neutral(white. That means damage to the hot that must be repaired. Could mean part of the cable needs to be replaced. Can't tell without on site investigation.

...adding that a nick involving only the neutral can definitely trip a GFCI or AFCI breaker without involving the black (hot) wire at all. (Assuming there was some load on the circuit at the time, a neutral-to-ground fault might be detected by the breaker because the fault could create a secondary grounded path.)

Thus, it depends on the type of breaker. If this is an "ordinary" (i.e., non-GFCI/AFCI) breaker, I agree that it's likely the black (hot) was also compromised and a thorough inspection is in order.

"...accidently nicked the white electrical wire with a nail..."

Exactly what do you Mean by "white electrical wire"?

The jacket around a piece of plastic covered cable or the actual wire inside the cable?

If you just nicked the jacket no real issue.

If you nicked the white insulation it can be tapped up, but the cable should really be replaced.

If you nicked the copper conductor the cable must be replaced.

I don't know that much about electrical wiring so excuse my lack of knowledge on the subject. The wire that I nicked was a white wire. I'm assuming that this is a neutral wire. The jacket was nicked. I can see some wiring though but doesn't appear to have damaged anything. What would I see if I damaged the copper conductor? Would I see any side effect to power other than the obvious fire danger?

I don't know that much about electrical wiring so excuse my lack of knowledge on the subject. The wire that I nicked was a white wire. I'm assuming that this is a neutral wire. The jacket was nicked. I can see some wiring though but doesn't appear to have damaged anything. What would I see if I damaged the copper conductor? Would I see any side effect to power other than the obvious fire danger?

Are you talking about a white conductor in a cable or conduit? Or are you talking about a white cable with multiple conductors in it?

I'm not sure. The wire itself says ground wire. I don't think that the actual wires were damaged, but I'm not sure that is why I'm asking. All outlets and lights associated seem to be working. Would there be any signs of problems?

rahatch-
Could you post a picture of the wire /cable you nicked to clarify for everyone?
That would assist in diagnosing the problem/solution for you.

>The wire itself says ground wire

What exactly does it say? I never saw a wire with the word ground printed on it in house wiring.

It appears you are using the word 'wire' for what the pros would call 'cable'. A cable has a sheath around several wires. It would likely be a half-inch or so wide and say something like Type NM 14 AWG (or 12) w/Ground and some other stuff.

If a nail through a cable caused a breaker trip, you almost certainly nicked both the hot wire and the internal (bare) ground wire.

>If the breaker tripped you hit the hot(black) wire and...

Unless the white wire he hit (and likely grounded out) was actually a switch loop hot?

The wires SHOULD not have been close enough to the roof to be hit by a nails.
I would open each device on the circuit and look for loose wires. Like someone already stated it is likely the hammering has knocked a wire nut off and the wires are shorting to the box.
Check the easy stuff first.