Rosenthal, who replaces Joel Hyatt, has served on Current's board since it was founded four years ago.
Hyatt will remain with Current in a vice-chairman role.
Last year, Current Media announced plans to make 30 staff redundant.
And earlier this year, Current Media cancelled plans to list on the New York Stock Exchange. Current Media, broadcaster of the user-generated TV channel Current TV, said it decided not to proceed with its public offering "in light of current market conditions".
Current Media said Current TV is watched in more than 51 million households in the US and UK, where it is distributed by Sky and Virgin Media.
Gore served four terms in the US House of Representatives and two in the Senate. He served as vice-president to Bill Clinton and was defeated by George Bush in the disputed 2000 presidential election. He has since become well known for his environmental activism.
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