First 2 EJ's of my career

[Thaal]

Every coach/ manager knows, or should know, that one umpire can not over rule another.
In stating that fact in your reply you are inferring you disagree with the call.
Backing up your partner correctly indicates you are a team working together.
Your intension may have been to defuse the situation, but the result, as you stated, was very different.
If you had directed the manager to you partner, being the guy that made the call, he could have come to you for help, or dealt with the situation himself. You said latter that you don’t know what your partner saw, give him a chance to explain it. If you need to step in to help out an inexperience umpire, do it when he needs help.