As far as I understand, impeachment is, by definition, removal from office. Why? Because criminal charges cannot be brought against a President while he is in office (just as there are certain indemnities for Congressmen/women, etc)

According to Britannica, for example:

In Great Britain conviction on an impeachment has resulted in fine and imprisonment and even in execution, whereas in the United States the penalties extend no further than removal and disqualification from office.

So in the US, if the person is not in office, they cannot, by definition be impeached. In the US, the entire purpose of impeachment is to remove from office. It's after that all the fun starts.

Also:

https://www.merriam-webster.com/dictionary/impeach

to charge (a public official) with a crime done while in office

So, you see, Obama is NOT the President of the United States. He was, now he's not. So he cannot be impeached - cannot be removed from an office he does not hold.

However, any former President could definitely be brought up on charges. They are civilians. They have perks, etc, but they are NOT office holders in the US govt.