You are a new manager -- congratulations!
Good for you for asking for help with this. It's one of the hardest parts of being a manager.
This member of your team is, either unconsciously or consciously, challenging your authority. He is testing the limits. He may or may not realize it, but he is trying to see whether he can get you to break. We've all done that sort of testing, since our parents taught us to put on our own clothing. It may seem childish, but it's human. It happens all the time.
Authority, in business, is a job, just like knowing the best way to do some technical task is a job and persuading a customer to buy your product is a job. You have this job because someone in your company decided you have the judgement and wisdom to use authority wisely.
From your question it seems that you deserve this trust. You're not taking this personally. Well done! You need to deal with this situation politely and firmly. It's not about you, it's about the strength of your whole team.
First of all, ask for help from a fellow manager. Don't try to do this alone. You're a new manager, and this is the kind of hard challenge all managers get. But it's new managers that have the hardest time with it.
Go to your boss, or your mentor, or somebody in human resources. Say "I need your advice. Person X is engaged in testing my authority by 'spreading bad ideas about me' (use your words). This is undermining my ability to lead the team. What can I do to intervene to stop this behavior?" Your boss may be able to either help you intervene or give you a strategy for intervening.
Here's a suggestion for a conversation with this Person X. Take notes in a couple of incidents where he says things about you that test you, so you have his exact words.
Then, have a private conversation. Say something like this. "I have something to say, and I want you to listen until I am finished speaking. When you said 'whatever he said' in public, I understood you to be challenging my authority. When you challenge my authority in public, you don't just challenge me but you challenge the success of our whole team. That's bad for you, bad for me, and bad for us all. Please do not do this in future."
He is not going to smack his forehead and say, "oh, you are right! what was I thinking? I will not do this any more."
Instead, he's going to try to argue with you. Do not engage him in a discussion of what he said or his motives. He probably doesn't really understand them himself. Let him rant for a while. Just listen. Don't react. It is very hard to keep quiet when a person is saying the kinds of things he will say, but you still are very wise to keep quiet. Be like a duck in a rainstorm: let the water roll off you.
At the end of the conversation, say to him "thanks for hearing me out. I know you'll think about what I have said." Then decisively end the conversation.
Good luck. This isn't easy, but you can do it!
One other thing to keep in mind: nobody is indispensable. Strong teams are always more effective in the long run than superstar individuals.