The German announced his retirement on Friday, five days after clinching his first world title ahead of Hamilton.
Hamilton said: "It doesn't matter who they put alongside me. I'll race them. I'm here to race. The focus is on next year and hoping for a better season."
Hamilton added that Rosberg's decision was "not a surprise" to him.
"This is the first time he has won in 18 years [of racing together]," Hamilton said.
"He has a family he wants to focus on and F1 takes so much from you.
"It is definitely going to be strange and it will be sad to not have him in the team next year."
Hamilton said he wanted Mercedes to pick someone who would work hard for the team.
"I've never needed a team-mate to be able to push me," he said. "The better the driver, the better it is for me when I finish ahead of them.
"You need someone with good values coming in and representing the team in a positive way, good work ethic and consistency, obviously, so we can those win constructors' championships.
"So, in terms of challenging for constructors' titles, which is obviously the team's first goal, we need to make sure we have someone equally as consistent as what we had before."
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But beyond that, he said, he would make no major requests of team boss Toto Wolff, but would offer his opinion if it was requested.
"If they want to talk to me, they can talk to me about it," Hamilton said.
"It should be someone who has to work well in the team. There will be stronger characters and less strong characters - and some drivers who take alternative routes in order to deal with competition.
"There will be good and bad ones out there, for sure - I would be more than happy to help with that analysis.
"I just always asked to have equal rights. As long we're treated fairly, it doesn't really matter who is alongside you.
"Of course, we've got great team bosses who I'm sure will choose the right people to be representing the brand."
He added: "I'm interested to see who would want to be my team-mate - and respect anyone that would want to."
Lewis Hamilton says he "doesn't care" who Mercedes pick as his team-mate next year following Nico Rosberg's decision to retire from Formula 1.