Bryant engaged in separate sideline confrontations with the duo during his side's dramatic 31-30 defeat to the Detroit Lions on Sunday, although he claimed that his actions were down to competitive spirit rather than anger with either player.
The 24-year-old reiterated that stance a day after the game and stressed that he had nothing but love and respect for his teammates.
"Tony Romo and Jason Witten, those are our leaders," Bryant told reporters. "Those are our main guys. The reason why I'm able to voice my opinion is because those guys helped me become who I am to where I can do that.
"Whenever I do things, I promise you all, man, it's all out of trying to keep our team motivated, keeping the passion, letting people know this game is important to us. The relationship me, Tony and Witt have, it's huge. Our relationship is great. On and off the field, it's outstanding.
"I think what most people don't get is everything isn't supposed to be peaches and cream during games. There's always something that doesn't go right, the way you think it should. It's the nature of football, period."
Bryant added that his teammates are like brothers to him.
"As far as our relationship, Witt did nothing wrong to me," he said. "Tony did nothing wrong to me.
"Witt loves me like if I was his little brother, Tony loves me like if I was his little brother and I love them for just being in my life, helping me change a lot in my life, helping me see different things about myself that I didn't see."
The Cowboys are currently top of the NFC East and will face the Minnesota Vikings on Sunday.