Kantner, who also was kidnapped by Somali pirates years ago, tearfully spoke about the militant threat and the February 26 ransom deadline in a video circulated by the SITE Intelligence Group, which monitors jihadi websites.
"Now they'll kill me", the 70-year-old man said before he was executed on Sunday after a ransom demand deadline passed.
Jurgen Kantner was abducted from his yacht off Malaysia in November.
The Abu Sayyaf is also believed to be holding a Dutch bird-watcher kidnapped in 2012, and has been blamed for abducting 18 Indonesian and Malaysian sailors from tugboats near the southern Philippines over the past month.
Kantner and his partner, Sabine Merz, were taken hostage by Somali pirates in 2008 and later freed, officials said.
They have frequently killed hostages if their demands are not met, and previous year murdered two Canadians.
Local media reports said talks had been conducted with the militants over Mr Kantner's release in exchange for 30 million peso (£500,000), but that they later broke down.
Government envoy Jesus Dureza confirmed the killing.
Since 1991, the Abu Sayyaf - armed with mostly improvised explosive devices, mortars and automatic rifles - has carried out bombings, kidnappings, assassinations and extortions in a self-determined fight for an independent province in the Philippines.
"I know it's risky sailing off into Somali waters and I have no private security guarding me, but I pray to God that pirates won't get me again". They were asking for P30 million in exchange for the safe release of Kantner.
Duterte has ordered the military to launch an all-out offensive against the estimated 500 Abu Sayyaf bandits operating mostly in the hinterlands of southern Philippine provinces of Sulu and Basilan.
Kanter was abducted from his yacht, the Rockall, off the southern Philippines previous year.
It was unclear why Ms Merz had been killed.
After he was rescued, he returned to Somaliland to collect his yacht, saying: "My boat is my life and I don't want to lose her. I don't care about pirates and governments", he told reporters from AFP in 2009.