Former Leeds, Everton, Newcastle, Bolton and Sheffield United midfielder Speed was boss of the Welsh national team - for whom he had also played - when he died in 2011 aged 42.
Ramsey, captain of the side under Speed, wrote on Twitter on Friday: "Remembering Gary Speed today.
"I know he'd be proud of what the Welsh team have achieved so far in getting to France."
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Speed is widely credited with having laid the foundations for the success Wales have enjoyed under his successor Chris Coleman, who has guided them to qualification for next summer's European Championships in France - their first major finals in 58 years.
The team won five of their last seven games under Speed, whose final match in charge was a 4-1 friendly win in Cardiff over Norway on November 12, 2011. Just 15 days later the Football Association of Wales announced he had died.
Arsenal star Ramsey's message was one of a host posted on Friday in tribute to Speed.
The Welsh FA said on its official Twitter feed: "As we remember Gary Speed, qualification is a fitting tribute to his memory."
A statement on the Welsh FA's website had already said it would be marking the anniversary by illuminating Speed's image in the window of its offices in Cardiff, and that staff would place flowers at the entrance and share a moment of quiet reflection.
There was also a message on the official Twitter feed simply saying: "#RIP Gary Speed."
Leeds, with whom Speed won the First Division title in 1992, wrote on their Twitter feed: "Four years on from his tragic passing, today we remember the great Gary Speed. Never forgotten. RIP Gary Speed, one of our greatest."
Everton added: "Four years ago today, the football word lost a true great. RIP Gary Speed, gone but not forgotten".
A message from Newcastle said: "Four years ago today, the football family lost one of its favourite sons. In memory of Gary Speed."
Bolton and Sheffield United also posted tribute messages.
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