Can we say that the word "wholeheartedly" means the same as the expression "without a shadow of a doubt"?
They can have similar meaning but not always.
They both indicate totality, but:
"Wholeheartedly" expresses enthusiasm.
"Without a shadow of a doubt" expresses confidence in an idea or statement (and suggests that there could have been doubt).
If you believe something "beyond the shadow of a doubt" then you also agree with it "wholeheartedly." In this sense, there is overlapping meaning.
However, you can do something wholeheartedly (with enthusiasm) that it would be incorrect to say "without a shadow of a doubt."
He put himself to the task wholeheartedly. (He enthusiastically did the task).
He put himself to the task without a shadow of a doubt. (He did the task without any doubt that he should do it).
Without a shadow of a doubt, he put himself to the task. (There is no question that he did the task.)
*Note how, by placing the adverbial clause at the front of the sentence, the meaning can change.
It is a subtle difference, but the person in question could do the task unenthusiastically but still knowing he absolutely should do it (cleaning the toilet, for example).
In addition to the nice answer from @rpeinhardt, note that wholeheartedly always applies to the person doing the action:
Bob wholeheartedly agreed.
Without/beyond a shadow of a doubt can be used to describe the inner conviction of the observer of the action:
The bloody knife proves, without a shadow of a doubt, that the murder happened in this room.
Here, it's not the subject of the sentence (the knife) that has no doubt, it is the unmentioned observers of the knife (perhaps the police) who now have no doubt. You would not use "wholeheartedly" in this sentence because a knife has no ability to be "wholehearted."