Austrian Thiem, who had received a first round bye as the fifth seed, lost 4-6, 6-4, 7-5, in the final warm-up tournament before the start of Roland Garros on May 26.
Verdasco will play Russian Karen Khachanov, the 13th seed, later in the day for a place in the quarter-finals, as competitors double up after play on Wednesday was wiped out because of rain.
"I really dislike how we players get treated at this tournament because yesterday was, in my opinion, not acceptable," said Thiem.
"They knew that not one ball of tennis was going to be played, and they hold us here until 7.30pm."
He said players were then caught up in clashes between fans before the Coppa Italia football final in the nearby Stadio Olimpico.
"They also know since, I don't know how long, that there's going to be the football match last night," continued the 25-year-old.
"The ride back from here to the hotel took like one and a half hours.
"If everybody knows that it's going to rain all day, then let the players leave at 2pm or 3pm when there's no traffic, when there are no guys beating each other up on the street.
"It's no problem if it's raining. It's also no problem if we have two or three matches a day.
"It happens from time to time. But then the tournament should also help us a little bit.
"I mean, I was having dinner. I couldn't even have treatment, otherwise I wouldn't get enough sleep. Then they schedule me at 10am today. I don't think that's acceptable.
"I'm not just talking just me. They took (Denis) Shapovalov and (Novak) Djokovic here!" he added, of a match which had been scheduled to play late on Wednesday.
Thiem, the winner of the Barcelona title last month, ended Roger Federer's return to clay in the quarter-finals in Madrid before losing to Djokovic.
"Of course, it's one of the reasons why I lost today," he continued.
"The other thing is that I arrived late on Sunday (from Madrid).
"I'm quite pissed about it.
"I was tired, exhausted, today because of all these shitty things," added Thiem, who defeated Federer in the Indian Wells final in March.