The contest is now set to take place on April 20, with the venue remaining Wembley Arena.
Nathan Cleverly’s WBO light heavyweight bout with Robin Krasniqi will also be delayed and kept on the same bill, and the undercard is unchanged.
Vazquez’s virus had been caused by food poisoning and he was advised to rest and rehydrate before resuming his training programme.
“I’m very sorry about this situation," said Vazquez.
"We were at the height of the preparation that was so intense my body was in a great physical condition, strong and fast. There were weeks of total dedication and preparation, but the infection I suffered in the stomach was very strong.
"I even laid down in bed for a week…but it was the best thing. My company told me that the fight will be rescheduled and in a few days more they’ll tell me the date.”
In an official statement Frank Warren Promotions said that while a stand-in opponent had been considered, “at this stage of his career, it was essential that the unification contest went ahead, even if it meant a slight delay.”
Promoter Warren added: “It’s very unfortunate that Vazquez has had to withdraw so close to the fight, but it is on medical grounds, which is beyond my control, due to his viral infection, and we have his doctor’s certificate confirming this.
"Therefore I have had to postpone the fight, not cancelled as stated in some media reports, to Saturday 20th April at Wembley Arena. Everybody is disappointed by this postponement, but we have worked hard to ensure that Ricky’s unification fight still goes ahead.”
On reports that Vazquez was fine, and had withdrawn for other reasons, Warren said: “Over the last few days there have been false rumours and inaccurate information circulating as to why Vazquez withdrew from the Burns fight.
"Yesterday I released the doctor's certificate to confirm that Vazquez had suffered a viral infection and now Vazquez has released a statement to confirm this.
"I hope that this now puts an end to the matter once and for all and we can focus on the new date of Saturday 20th April. I was as disappointed as the fans when I had to postpone, but the most important thing is that the show is still on, Ricky still has his unification fight against Vazquez and Cleverly still defends against Krasniqi.
"It’s a tough fight for Ricky and as respected journalist Graham Houston said in the latest edition of Boxing Monthly ‘Ricky Burns and Miguel Vazquez is a genuine fight of the highest order’.
"Cleverly also has a very hard fight, but if wins he can then look towards a showdown with the winner of Tavoris Cloud and Bernard Hopkins this Saturday.”