The singer, 48, and Joshua Sasse announced their engagement in February after a whirlwind romance, and both were forced to deny they had wed last month after a report claimed they had tied the knot in Greece on Minogue's 48th birthday.
28-year-old actor Joshua posted a poem entitled "To My Wife" by William Ernest Henley on Instagram.
"Take, dear, my little sheaf of songs, For, old or new, All that is good in them belongs Only to you; And, singing as when all was young, They will recall Those others, lived but left unsung - The best of all," it reads.