They bundled the baby animal into their car and drove about 48km to the nearby town of Durango to drop it off at a local humane society shelter.
But because the humane society didn't handle wildlife, staff contacted wildlife officers who decided the "most humane thing to do was to euthanise it", Colorado Parks and Wildlife spokesman Joe Lewandowski said.
Mr Lewandowski said the tourists were wrong to assume the baby deer had been abandoned by its mother.
He said mother deer often left their young for extended periods of time while going off to collect food, and rarely abandoned their children.
But once it had been intercepted by humans and displaced, it would be difficult for the animal to return to the wild.
"Unfortunately, we had to euthanise this deer because turning it loose in the wild would be cruel, especially since we don't know where it had been taken from or where its mother is," Mr Lewandowski told ABC News.
"The mother would have had the nutrition needed to nurture the deer and keep it alive, and we don't have such nutritional products available."
Deer "also need to learn from other fellow deer how to eat, how to move across the landscape and how to escape predators," Mr Lewandowski said, adding wildlife officials would be unable to teach fawn such critical knowledge.
He urged the public to stay away from animals in the wild and call experts if help was needed.
"In many cases they can tell you what you should do over the phone, or they can even send out a biologist or wildlife officer who can then determine what the best approach is," he said.
"The best thing people can do is to look at wildlife, admire their beauty and move on.
"They've done quite well without us for thousands of years and will continue to do so."
The death of the fawn echoes the death of a baby bison at Yellowstone National Park in May, which was put down after two tourists kidnapped it from the wild, thinking it "looked cold".
The father and son loaded the newborn calf into their car and took it to a ranger station, but when rangers returned it to where it belonged, it was rejected by its herd and had to be euthanised.
Authorities condemned the actions of tourists, who were branded "stupid", "idiotic" and "dumbasses" on social media.