Did an ancient Greek philosophy really inspire cognitive behavioral therapy?

Yes, by its founders' own account, though Stoicism was not its only ancestor. Stoic philosophy was the original philosophical inspiration for modern cognitive psychotherapy, particularly through Albert Ellis's rational emotive behavior therapy (REBT), the major precursor of cognitive behavioral therapy (CBT). The original cognitive therapy manual for depression, by Aaron Beck and colleagues, says that its philosophical origins "can be traced back to the Stoic philosophers".

Ellis's central principle was that people are rarely emotionally affected by external events but by their thinking about such events. In his first major book he wrote that it "was originally discovered and stated by the ancient Stoic philosophers." A quotation from Epictetus's Enchiridion, that it is not the events that upset us but our judgments about them, was taught to most clients in the first session of traditional REBT. REBT posits that people have erroneous beliefs about their situations, that these beliefs cause disturbance, and that they can be disputed and changed.

Beck came at it from a different direction. Working with depressed patients, he found streams of negative thoughts that seemed to arise spontaneously. He called them "automatic thoughts", and sorted them into negative ideas about oneself, the world and the future. Both men cited Stoicism as a forerunner, and Beck cited Epictetus in particular. But they were not copies of each other: Beck highlighted the Socratic method, Ellis highlighted disputation.

So the link is real, but it is a lineage of ideas, not a simple transplant. Precursors of REBT have also been identified in Confucius and the Buddha.