Your boss is half-right, which means he is right. To answer your question, it is recommended to look at the organizational structure you, your team and your boss are in. If there is no structure, call it chaos. Your boss puts you in the seat of a full-time (not just coordinative) PM, with considerable authority (responsibility). You might be in a functional organization, where each employee has one clear superior (the PM, you), unless your team members are for example colocated and there is maybe a organizational unit, i.e. department.
You must communicate the findings you have to senior management with a pinch of warning (risk). Additional, I would suggest you come up with a contingency plan. This is expected. Meanwhile, you are still accountable (or held) for the outcome of happenings and overall project success.
Based on the situation, your boss (what is your boss, a senior pm, sponsor or director?) recommends you to double check his work. Simply put, that is what you are going to do for a week. You then report that you have double the findings and negotiate with him the next strategy...
If you have not, read about the so called "Set-up-to-fail syndrome" and double check with your current situation.
Havard released "HBR's 10 MUST READS" series, here "On Managing People". Literally all chapters are good, but Chapter on "Set-up-to-fail syndrome" and "Managing Your Boss" can make a difference for you in the next weeks, albeit the title "On Managing People" is just a percentage of the PM pie.
Another last point. Are you sure that you reacted with a relaxed, thoughtful attitude toward your colleagues mistake? Form your description it does not look like it. A mistake is a mistake, if it was a mistake. On top of that, you have not made it. Who reported to your boss then, you? Or did your boss had the same finding? Now, how happy is your colleague with being known (and tracked)? He is going to be really scared, maybe. Overall, the fact that you wrote "you manage him", call him "he is error prone" and "errors he makes are quite simple." makes me doubt your qualities.
PMBOK 5th, PMI
Keywords: PMBOK, Functional organization, Matrix organization, Set-up-to-fail syndrome, Micromanagement