Coleman will leave hospital after surgery for a double break, suffered in a tackle that saw Neil Taylor sent off.
Williams also revealed Taylor wanted to see Coleman in hospital and sent him a text message from the Aviva Stadium.
"Neil was gutted. He was sitting on the floor virtually in tears," he said.
Williams added in an interview with the Everton website: "Even people who don't know Seamus like I do were gutted.
"He asked me to check if it would be OK to go to see Seamus in hospital. He got his number off me and sent him a text straight away.
"When I found out it was a broken leg it made me feel sick inside because it's someone who I like so much and is one of my mates.
"He's going to have up and down days but the way he is, I'm sure his determination will help with his rehab and hopefully we'll have him back sooner rather than later and stronger than ever."
Williams says Coleman has become one of his best friends at the Merseyside club since joining from Swansea City last summer.
The centre-back says he did not initially realise the extent of the injury, which was so severe that broadcasters refused to show replays of the incident.
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"Seamus is one of my best mates at Everton, someone who I've got really close to," Williams said.
"For me personally, I didn't really know anyone at Everton when I came to the club but he was someone who made me feel welcome and helped me to settle in straight away.
"I tried to go to the hospital Seamus was in after the game but the FAI (Football Association of Wales) said he'd had a few family visitors and they were trying to get him to sleep.
"I sent him a text and he woke up in the night and texted me back. I spoke to him then. I've spoken to him since, too, and he seems positive and focused on his recovery."
Williams was pictured comforting Coleman as he was taken off the field on a stretcher and says he initially had not realised the extent of the damage.
"I didn't really see the tackle. All I saw was the red card and my initial reaction was to go to the referee," he said.
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"At the time I was in the swing of things. It took me a minute to realise, 'Actually, he hasn't got up,' and that it was Seamus who was down.
"At that point, I wasn't thinking about the game, or us being down to 10 men, I started thinking, 'Is my mate okay?' That was when I went over to him.
"Even at that point I didn't know it was so serious. After the game I went straight into their dressing room to speak to James McCarthy and he told me Seamus had gone to hospital and had broken his leg.
"It was just a weird feeling from everyone around the dressing rooms."
Wales captain Ashley Williams says he "felt sick inside" when he realised Everton team-mate Seamus Coleman had broken his leg in Friday's World Cup qualifier in Dublin.