The answer is yes depending on which state you live in.
First off this is not the .177 Daisy you are used to. The rifle featured in the video below is an AirForce Texan. The airgun is a a PCP or pre-charged pneumatic rifle that shoots .45 caliber slugs which are the same bullets used to load .45 Colt rounds. The AirForce Texan also produces over 500ft-lbs of kinetic energy at the muzzle which is about the same as the .45 Colt or a .45 ACP. Plenty of people hunt deer and turkey with a .45 ACP so you should be able to use this airgun to harvest the same animals depending on your local rules and regulations.
There are currently only a handful of states that allow hunters to harvest whitetail deer. Those states include Alabama, Missouri, Arizona, Virginia, and Michigan. Hunters can also use airguns to harvest turkeys in Vermont, California, Michigan, Virginia, and Maryland. Laws and regulations are constantly changing so check with your state wildlife agency before you decid to go hunting with a big-bore air gun.