If you are asking purely about syntax, replacing を with も is not a problem at all.
However, your sentence has several other issues.
First of all,
あなたを気にしない is not quite idiomatic.
あなたのことを気にしない sounds much more natural. Besides, or perhaps because of this reason,
あなたも気にしない would most likely be understood as:
You don’t care, either.
俺は should be
俺が. This は takes out 俺 as a topic, but a topic usually doesn’t go inside a subordinate clause.
After these corrections (and a comma for readability), we arrive at the following:
Though this sounds natural enough, I would prefer:
Let’s agree on this as the base sentence to which to add the meaning of も, for the sake of the following discussion.
Now, I understand what you want to say is:
If I were him, I wouldn’t care about you, either.
However, there is a little ambiguity to this English sentence because it could be interpreted in, at least, two ways:
- He doesn’t care about you. If I were him, I wouldn’t care about you, either. [i.e. I am like him.]
- He may not care about other people, but he cares about you. If I were him, I wouldn’t care about you, either. [i.e. You are like other people (for me).]
This ambiguity needs to be resolved if the above sentence is to be translated to Japanese.
Your approach of replacing を with も would result:
This works only in the second scenario.
In the first scenario, も would have to go with 俺 because that's what you want to put into the same group as something else, which is the main function of も. One straightforward solution would be to make the subject of the main clause, which is 俺, explicit and add も to it, like this.
However, this sounds not only redundant but also a bit unnatural because
俺が彼なら somewhat suggests what follows will talk about some contrast between “me” and “him”, yet the main clause contradicts it.
It would be more natural to say:
This would be interpreted as meaning either one of the following:
- He doesn’t care about you. In his shoes, I also wouldn’t care about you.
- I don’t care about you. Even in his shoes, I still wouldn’t care about you.
Some ambiguity remains, but context will take care of the rest.