Mr Poots lasted just three weeks in the job after he ousted Arlene Foster from the position.
A formal process will now be launched by the DUP to find a successor.
The front runner for the job is Lagan Valley MP Sir Jeffrey Donaldson who was defeated last month by Poots in the first leadership race of the party’s 50 year history.
Said the Foyle MP: “Politics is very difficult, of course I feel personally sorry for Edwin Poots.
“But there is a lesson here for unionist leaders and for the next unionist leader if it is Jeffrey (Donaldson) or not.
“That I don’t think there is anything good will come from the type of politics that is being played out by unionism over the past number of months.
“You have to sit down with the rest of us, work through these issues.
“Creating crises, threatening to walk away will get you absolutely nowhere,” added Mr Eastwood.
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