The versatile Alaba, who arrives on a free transfer from Bayern Munich, will wear the number four shirt that Ramos sported during his 16-year stint at the Santiago Bernabeu.
Ramos, 35, has joined Paris Saint-Germain.
Alaba insisted that he does not feel any pressure to step into the Spaniard's shoes.
"We all know Ramos was here for a long time and wore this number, becoming the leader of this team," he told a news conference.
"But the truth is that yesterday I spoke to the club and they told me that right now it's the only available squad number, and I didn't want to start asking other players if they wanted to swap.
"I'm honoured to wear the number four, it's a source of great pride and motivation for me and I want to do it justice - but I'm not here to compare myself to other players or replace them: I'm here to be David Alaba."
The 29-year-old Austrian has signed a five-year deal with the 13-time European champions, having previously spent his entire senior career with Bayern where he won 10 Bundesligas, two Champions Leagues and six German Cups.