He was born in Campbelltown in Tasmania to Methodist minister Henry Greenwood and Caroline Jane Tuckfield.
The family moved to Victoria around 1890, and Greenwood became an office boy and from 1897 a farm labourer.
From 1904 he ran a tent manufacturing firm, which eventually expanded to become a large softgoods warehouse.
In 1917 he was elected to the Victorian Legislative Assembly for Boroondara; he was considered a Nationalist, but never sought formal party endorsement.
He continued in the Assembly, transferring to Nunawading in 1927, until he retired to allow Robert Menzies to run for a lower house seat in 1929.