Mercedes' Hamilton secured victory in the British Grand Prix after the lap-one contact with Verstappen to cut the Red Bull driver's championship lead to just eight points
The duo had previously gone wheel to wheel on a number of occasions this year but always maintained a healthy respect for one another.
But after what unfolded at Silverstone, former F1 champion Button believes the blue touchpaper has been lit.
"It's going to be a feisty one from here on, which is good for us watching but whether it's good for them, I don't know," Button told Sky Sports.
"Emotions are running so high. These are two of the best drivers that have ever driven F1 cars and it is going to be fisticuffs over the next few races."
After Verstappen was taken to hospital for "precautionary tests" following his 51g impact, Hamilton came under fire from numerous figures inside Red Bull, notably team principal Christian Horner who accused the Briton of being "dangerous".
Button is hoping emotions do not run too high, starting with this weekend's Hungarian Grand Prix.
"We've just got to hope it's done in the right manner," he added.
"I think after this race [Hungary] hopefully they'll discuss it. That's what's important to me, it's that these drivers understand what happened and they talk about it and they move on, and then they can fight as they should with the gloves off."
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