The world number three marked his return to Wimbledon after last year's ban with a straight sets win over French-born British wild card Arthur Fery.
Medvedev admitted he was unsure how British fans would react to the return of Russian and Belarusian competitors, who were banished from Wimbledon in 2022 in response to the war in Ukraine.
"The reception today, I don't feel it that often. I was really touched by it," said the 27-year-old after his match on Court One.
"I'm going to try and give it back, show some good tennis.
"I will try and not get crazy, not be a selfish kid on the court. Sometimes I can be."
Former US Open champion Medvedev saw off the 391st-ranked Fery 7-5, 6-4, 6-3.
Medvedev enjoyed his best performance at Wimbledon on his last appearance in 2021 when he reached the last 16, defeating former runner-up Marin Cilic on the way.
The former world number one arrived in London with five titles to his name already this year -- four on hard courts and one on clay in Rome.
However, his confidence was dented by a shock first round loss at the French Open to Brazilian qualifier Thiago Seyboth Wild.
A run to the quarter-finals in Halle last month at least gave him a degree of confidence on grass.
"When I became a top guy, especially on hard courts, I felt like I can win a lot of matches just by kind of knowing what to do," he said.
"But at this moment I don't feel at home playing on grass. I have every opportunity to try to make it a good one, and I'm going to try to make it these two weeks."
Medvedev will face Adrian Mannarino of France for a place in the last 32.
