Crowd favourite Federer was untroubled in a 6-2 6-2 victory on Centre Court and said afterwards: 'I had a good time on court. I thought the atmosphere was good.'
The top seed is the reigning Olympic Doubles champion with Stanislas Wawrinka but, were he to win Singles gold, he would be following countryman Marc Rosset, the surprise winner at Barcelona 1992.
Federer, who next meets Uzbekistan's Denis Istomin, added: 'I definitely get inspired by the 1992 victory by Marc Rosset. That was huge news in Switzerland.
'But I'm still very far away from all the medals in Singles and Doubles. Let's not get carried away too quickly. I'm happy with my level of play.'
The USA player, who is unseeded here, came through a tight clash with Slovakia's Martin Klizan 7-5 6-4, while Australia's Lleyton Hewitt dug out a 6-3 4-6 6-3 victory over Ukraine's Sergiy Stakhovsky.
Meanwhile, seventh seed Janko Tipsarevic will face 10th seed John Isner in a tasty third-round clash after both won their matches today.
Another attractive clash in two days' time will see eighth seed Juan Martin Del Potro of Argentina face 12th-seeded Frenchman Gilles Simon after their respective victories over Andreas Seppi and Grigor Dimitrov.
Eleventh-seeded Spaniard Nicolas Almagro is also in the third round, where he will face Steve Darcis of Belgium, while in the first round there were wins for Richard Gasquet, Juan Monaco, Marin Cilic and Milos Raonic.
Ranked 302 in the world, Vardhan is playing Doubles with Leander Paes. They won their Doubles opener 7-6 (7/1) 4-6 6-2 against Dutch pair Robin Haase and Jean-Julien Roger.
Federer and Wawrinka opened their Doubles defence with victory over Japan's Kei Nishikori and Go Soeda, but the sixth seeds had to come from a set down to win 6-7 (5/7) 6-4 6-4.
There was disappointment for Novak Djokovic, though, as he and Serbian partner Viktor Troicki were beaten 7-6 (10/8) 6-3 by Swedes Johan Brunstrom and Robert Lindstedt.