The quick answer is no. The vast majority of the hundreds of thousands of TEFL-certified English teachers working around the globe do not possess a CELTA - they have a different brand of TEFL certification.
If you've previously heard or been told that you "need to have a CELTA", this is false. It's the equivalent of someone telling you that you can only earn a recognized college degree from one particular university.
At International TEFL Academy, we certify more than 6,000 students a year who gain an accredited TEFL certification - not a CELTA - and our graduates successfully gain employment in 80 countries worldwide.