Dixon finished just two-tenths of a second behind Arrow McLaren’s O’Ward, who punted him out of the Long Beach Grand Prix in April.
The six-time champion was livid with the move, which O’Ward was unrepentant over, and declared: “If that's how the series wants us to race, then I guess it's all gloves off from this point.”
Having chased down O’Ward in the closing laps, Dixon was just unable to get within range to make a move to grab the final spot on the podium.
When asked by USA Network if he was tempted to get physical, he replied: “That day will come! We’ll see. But I think there’s a time and a place for it.
“We were trying to get close and he had more OT [push-to-pass allocation] than us. For the most part, we tried to race as cleanly as possible.”
Dixon produced a sparkling drive from 23rd on the grid, after his team was forced to build up a new chassis following his bizarre collision with Will Power in second practice.
“Obviously one spot away from a podium, which would have been nice, so huge credit to the #9 crew and Honda as well,” said Dixon, who carved through the field on an alternate strategy after pitting early.
“It was definitely a day when you needed driveability to look after those Firestones and also fuel mileage.
“It got a little rough at the back there a couple of times, pit strategy went the way it should’ve [but] it would’ve helped us a lot if they hadn’t waited [to throw the yellow] on that [Romain] Grosjean crash – they keep doing that – because it would’ve benefitted us a lot.
“Off-line was kinda rough, especially later in the race where you kinda had to make some dives, it took three or four corners for your tyres to clean up again. But it also proved beneficial for me today as some went off in front of me.
“The tyres went surprisingly well, I think a lot of people had a downer on how the reds were going to be, but they did a helluva a job.”
Dixon also paid tribute to team-mate Alex Palou, who scored the team’s 250th overall race victory on Sunday.
“Congrats to the #10 car, he’s having a hell of a championship right now and definitely got a great lead going on,” said Dixon.
“It’s time to get to work to see if we all can start closing that gap.”
Their team boss Chip Ganassi added of the milestone victory: “It’s a big number, just a testament to a lot of people having a lot of blood, sweat and tears and money in those 250 wins.
“The great employees we have, the great partners we’ve had over the years that have contributed to every one of those, it’s just an incredible achievement and it’s all about the team.”
