The in-form German, who beat Roger Federer to win the ATP Halle tournament before Wimbledon, won 4-6 6-3 6-4 in just over two hours against the world number 12 and second seed.
Rosol, who beat Viktor Troicki in the opening round of the German tournament, lost the opening set in a tie-break before winning the next, only for Ramos to come through a 7-6(8) 4-6 6-3 winner.
Fourth seed Marin Cilic of Croatia also progressed, beating Federico Delbonis of Argentina 6-3 6-2.
Spain's Feliciano Lopez came through a hard-fought match against Jan Hajek to make it into the quarter-finals, where he will face Bellucci.
Fifth seed Lopez lost the opening set to the Czech before coming through to win 6-7(3) 6-1 6-4.
One first-round match was played on Wednesday, as seventh-seeded Colombian Santiago Giraldo beat Henri Laaksonen of Switzerland 6-2 6-2.
Russell booked his place in the last eight with a 6-7(5) 6-4 6-4 upset of fifth seed Kevin Anderson of South Africa.
In the other quarter-final that was decided on Wednesday, Kei Nishikori will meet Go Soeda in an all-Japanese clash.
Nishikori downed Ricardas Bernakis of Lithunia 6-4 6-7 6-4 while eighth seed Soeda saw off Igor Kunitsyn of Russia 6-1 7-5.