Wissanu said his visit was routine after the envoy had already called on Foreign Minister Don Pramudwinai.
Davies, who assumed his official duties this month, described his meeting with Wissanu as good.
Their discussion focused on strengthening bilateral relations and cooperation in many fields including public health, law enforcement and two-way investment.
The talks also touched on political developments in Thailand, he said.
Wissanu also described the get-together as good.
Davies appeared to have done his homework thoroughly before taking up the post in Thailand.
“The ambassador can speak Thai and has a sense of humour,” he said.
The United States has scaled down its dealings with Thailand since the military coup in May of last year that was led by General Prayut Chan-o-cha, now prime minister.
The US has called for Thailand to restore democracy while suspending a series of cooperation and assistance programmes.
Prayut told reporters that Davies had asked to see him but their schedules did not match, so the envoy met the deputy prime minister first.
It was a matter of protocol for the US diplomat to make the rounds with other ministers first, Prayut said. That way the deputy PM could help explain things to the ambassador, who might not listen to what Prayut had to say.
He appeared testy when asked when the appointment with Davies was, while declining to reply. “Why, are we their colony? We give importance to every country,” Prayut said.