You can't worry about his relationship with your boss being friendly.
I find your disdain for office politics irrelevant, self-righteous and irritating considering the situation that you are in. Drop the attitude because your disdain may very well work to your detriment.
Give your boss the fact based version of what actually happened. That the call to you happened while he was running a personal errand. That when you called him for advice/support, he recommended that you reboot a server. Say that you were unwilling to reboot the server given that you were unable to reach anyone within the department to coordinate the rebooting of the server with. You eventually reached the hospital supervisor who stated that they were "taking point" on this issue, you logged the incident and left it at that. If simply stating the facts amounts to office politics to you, then there is nothing we can do for you.
Having said that:
I am wondering why you did not notify your boss nor did you follow up with your colleague when it was clear to you that you couldn't reach anyone in the department to coordinate the rebooting of the server with. You are not mentioning what you found when you checked the server logs for two hours but it looks like you found nothing. Your boss and your colleague should have known that you were unable to coordinate the rebooting of the server within say 30 to 60 minutes of you reacting to the incident.
You should have gotten back to your boss and your colleague when the hospital said that they were "taking point" on the incident. Because the incident was clearly unresolved from a technical point of view since the server had yet to be rebooted.
I don't know what your colleague did to resolve the incident but my guess is that he simply rebooted the server on his own and reported or in your words "tattled" on you to your boss afterward. The fact that you use the word "tattled" indicates that you are the petty party not necessarily your colleague. The fact is that the incident took place and that your colleague not you reported the incident to your boss - that counts as a mark against you. And you missed at least two opportunities to escalate the incident to your boss before your colleague did.
Your inability to coordinate the rebooting of the server with the department issue was a management issue and you should have escalated the issue to your management i.e. your boss the minute you realized that you were hitting the wall when you tried to go through the proper channels. In fact, both your boss and your colleague had the right to know and the concomitant responsibility to intervene when you realized that going through the proper channels was getting you nowhere.
You need to know when to resolve issues at your level and when to escalate. And you have to keep lines of communications with your boss and your colleague at all times. And make sure to report your colleague if he gets abusive or he fails to be responsive whenever you get back to him on anything.
Your concern that he is on friendly terms with your boss - that's irrelevant. This is the wrong time to worry about office politics. He probably knows enough to be on the boss's good side and for all anyone knows, he gets on the boss's good side by kissing up to him. Unlike you, he knows to keep the lines of communication open with his boss. You have to keep open your own lines of communication with your boss, too.
EDIT (in response to your edit):
Since you say that the root cause turned out to be - different? - server running out of disk space, no amount of rebooting of the server you were working on would have resolved the issue, unless you were working the transaction server, and the transaction server was running out of disk space because it was stuffed with temporary files that a reboot would have cleared.
The finding is clear. It is also clear that the responsibility for monitoring disk space on the servers is his. You don't have to say anything but state or restate the facts. How the boss deals with him is not your problem. I don't get why "I desperately want to avoid throwing him under a bus", it's a straightforward management issue that gets dealt with at your boss's level and your "snitches get stitches" attitude is neither relevant nor helpful to resolving the situation at hand nor to preventing the situation from occurring again.
The process of management cannot be effective if the accountability is not there. And if accountability is not laid under the right door, how do you expect the relevant party to be made to take the right corrective action? Which is preferable to the workflow of your department, that the right corrective action be taken or no corrective action be taken because you interfered with accountability being laid at the appropriate door because of your - highly judgemental - perception that you are "tattling" on him.
It is highly unprofessional to unnecessarily to inject "office politics" and "tattling" into a situation that can only be managed and resolved if the correct set of facts are on display and at hand and accountability for corrective action is laid at the appropriate door. I suggest that you review and revise your definitions of "office politics" and "tattling" - you'll be much happier and you'll be more effective that way.