He was born on the island of Bequia, Saint Vincent, and received a colonial primary school education. In 1930, Palmer received a scholarship to attend secondary school in London. With additional scholarships, he went on to study medicine at the University of Birmingham, graduating in 1941.
He returned to the West Indies and worked in Jamaica, Trinidad and Tobago and Saint Lucia. In 1950, Palmer returned to Saint Vincent to practice. In 1954, Palmer co-founded the People's Political Party (PPP) and was elected as a legislator to the Legislative Council. He was elected as Chief Minister in 1967.
In 1979, Palmer led Saint Vincent and the Grenadines to independence from the United Kingdom. He was knighted by Queen Elizabeth II the same year.
In 1984, Palmer retired from politics and was succeeded as Prime Minister by James Fitz-Allen Mitchell. He died on his 92nd birthday.