Paltrow, 48, cropped most of her own face out of the picture, leaving only Falchuk in the frame.
The couple, who met on the set of musical drama series Glee in 2010, tied the knot in 2018 in a ceremony at The Hamptons in New York.
Oscar-winner Paltrow, who has two children with ex-husband Chris Martin of Coldplay, was criticised by an NHS medical director last week for sharing her method for tackling Covid.
The actress and lifestyle guru said she was suffering from long Covid and had been left with “long-tail fatigue and brain fog”.
She said she has since embarked on a “keto and plant-based” regime, with no sugar and alcohol, and fasts until 11am every day.
Professor Stephen Powis, national medical director for the NHS in England, said that “serious science” should be applied and “influencers” have a responsibility to the public.
