Born in Plymouth, Massachusetts, Jackson studied medicine at Harvard and continued his education at the Sorbonne in Paris, working on chemistry and geology. He returned to America and set up a practice in Boston. Jackson's professional career consisted of a series of spectacular claims to the work of others. These started on his homeward voyage and were to persist until he finally became insane in 1873.

While sailing from France to America in 1832 Jackson befriended a fellow American, the portrait painter Samuel Morse, with whom he discussed the possibilities of electric telegraphy. When Morse exhibited his telegraph to Congress in 1837 he found that he had to establish a right to his own invention against Jackson's claim that Morse had stolen it from him. It took Morse seven years to prove the validity of his claim.

In July 1844 Jackson recommended to William Morton, a young dentist lodging with him, that he should try treating his patients using ether, which was commonly used by medical students as a joke. Morton took up his suggestion and found it promising. He experimented on himself, gave up his practice to work on dosages and systems of inhalation, and introduced the anesthetic to the medical profession. Nothing was heard from Jackson until it was clear that money and fame were going to be awarded to someone. When Morton went to Congress to ask for compensation for yielding his patent to the US government he found some senators who took him for a thief. When he went to Paris in 1847 to lecture on his discovery he found that Jackson had already lodged a sealed envelope with the Académie claiming a priority going back to 1842. Committees were set up by governments, states, academies, and professional bodies but Jackson managed to so confuse the issue that when Morton collapsed and died in 1868 he was still fighting his claim and still penniless.

Jackson became obsessive about the discovery, ignored his other work, took to drink, and spent the last seven years of his life in a lunatic asylum. He even wrote a book on the subject, A Manual of Etherisation (1861). Curiously, both Morton and Jackson have monuments in the same cemetery, both proudly proclaiming their triumph in alleviating the misery of mankind.