Williams was devastated after abruptly withdrawing from a fourth-round clash with Maria Sharapova with pectoral soreness.
Scans on the area will determine when, and if, the American can return, plunging her Wimbledon participation into jeopardy.
Goolagong Cawley was at Roland Garros to watch Williams, 36, and expects the 23-time major winner to bounce back — despite claiming the former world No 1 returned too soon.
“Look at how eager she is to get back,” Goolagong Cawley said.
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“She’s a true champion and she’ll work hard and want to get back.
“I wish her all the best because she’s a fantastic player, I don’t know how much longer it will take for her to get back to her old self but I wish her all the best.
“It’s wonderful being a mother playing.”
Goolagong Cawley won the 1977 Australian Open, aged 26, seven months after having daughter Kelly.
Due to receive the International Tennis Federation’s highest accolade — the Philippe Chatrier Award at the world champions’ dinner — Goolagong Cawley said Williams simply needs more time.
“I was in my late twenties when I had Kelly, but Serena is older,” she said.
“I knew she went back too soon before. Her body is not conditioned to playing matches, just like me.
“She is a bit older, too, so she needed to take a little bit more time before she came back into playing tournaments.”
Williams paid the price of contesting six singles and doubles matches in seven days after just four matches in the previous 16 months following childbirth last year.
The American said tests in Paris would decide her future.
“So I’m going to get an MRI tomorrow. I’m going to stay here and see some of the doctors here, see as many specialists as I can,” she said.
“And I won’t know that until I get those results.
“There is a lot of theories. I have never had this before. I have never felt this in my life. Like, this is so painful.
“So I don’t really know how to manage it yet. Sadly, when you do have an injury that you have had before, you can kind of manage it.
“I have pretty much had every injury in the book. But this is a little different, and, yeah, I’m clueless as to what to do.
I’m just going to do what the doctor thinks I should do and get all the evaluations on it.”
The winner of 23 majors was due to face Maria Sharapova in a sellout Roland Garros clash, but the appointment was wiped just minutes before the players were due to walk on court.
“I’m beyond disappointed,” Williams said.
“I gave up so much, from time with my daughter to time with my family. I put everything on the court, all for this moment.
“So it’s really difficult to be in this situation, but I just always try to think positive and just think of the bigger picture and hopefully the next events and the rest of the year.
“And I have made every sacrifice that I could. So it’s extremely disappointing.
“But also, I made a promise to myself and to my coach and to my team that if I’m not at least 60 per cent or 50 per cent, then I probably shouldn’t play.
“The fact that I physically can’t serve at all is a good indication that maybe I should just go back to the drawing board and stay positive and try to get better and not get it to a point where it could be a lot worse.
“Physically I’m doing great. Again, it hasn’t been easy. I sacrificed so much to be at this event.
“I can only take solace in the fact I’m going to continue to get better.
“And I had such a wonderful performance in my first grand slam back. I just feel like it’s only going to do better.
“And I’m coming up on hopefully surfaces that are my absolute favourite to play on and that I do best on.”
In a bizarre twist, Garbine Muguruza advanced to a quarter-final clash with Sharapova after Lesia Tsurenko retired when trailing 0-2 with a leg injury.
Simona Halep and Angelique Kerber will clash in the other top-half quarter.
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