Specify

No, because it mentioned her child in one sentence, and his child in the other.
Also, in the first sentence, no specifics were made regarding who was to take care of the child.

The word "specified" in your sentence needs explanation.

In his will, he specified that he wanted me to take care of his child . . . .

Other options:

He chose me to take care of his child . . . .

He specifically chose me to take care of his child . . .

I forgot to write "me" in the first sentence, but according to your answer, this would be wrong however.
He specified me to take care of his child after his death .

No - it is not wrong.

The "me" does explain the sentence.

(Be sure to proofread all your questions that you post here, please)

Ok. I will. Thank you