Brown was shown a straight red card for kicking Neymar after bringing the Brazilian down in the second half of the Champions League clash at Parkhead.
Celtic were comfortably holding Barca before the 28-year-old's moment of petulance and went on to lose 1-0 but his club manager, Neil Lennon, criticised Neymar for his reaction and questioned whether there was enough in the flick of the boot to warrant a red card.
Strachan, who was working as a TV pundit on coverage of the game, refused to give his opinion on the incident but insisted he retained trust in Brown, who has captained Scotland in recent games.
Speaking at the announcement of his squad to face Croatia on October 15, Strachan, who bought Brown to Celtic in a £4.4million deal in 2007, said: "I'm not going to speak to him, no. I'll look forward to meeting him again because he always makes the world a cheerier or brighter place when I meet him.
"And I know fine well when I pick him that I have got a great captain and a great player whose career has come on leaps and bounds in the last two or three years."
Strachan named a similar squad for Scotland's final World Cup qualifier to the one which lost to Belgium and won in Macedonia last month.
But Shaun Maloney, who scored a spectacular late winner in Skopje, misses out with a back injury that has forced him out of Wigan's last four games.
Norwich full-back Steven Whittaker is suspended while Celtic winger James Forrest, who has missed recent action because of sciatica and a virus, is also absent for fitness reasons despite coming off the bench against Barcelona.
Strachan said: "He was only on for 20 minutes and that was as much as they could push out of him, so it would be unfair of us to push him as well.
"But if I get a phone call from James or Lenny or anybody at Celtic saying he's feeling good about himself, then that's the way his injury is working just now.
"It is very temperamental, the injury he has got just now, and we'll let them get on with it.
"It would be nice if they picked up the phone but, if they don't, we understand it totally. We have not got a problem with that at all.
"It's best we look after the young man's future because he can be a real star for Scotland and Celtic."
Derby midfielder Craig Bryson, Rangers left-back Lee Wallace and Birmingham winger Chris Burke retain their places after being handed late call-ups last month.
Hull pair Allan McGregor and George Boyd are in the squad following injury along with West Brom midfielders Graham Dorrans and James Morrison. St Mirren goalkeeper Craig Samson and Nottingham Forest forward Jamie Mackie drop out.
Steven Fletcher again misses out alongside fellow long-term absentees Darren Fletcher and Gary Caldwell after the Sunderland striker dislocated his shoulder shortly after returning from an ankle problem.
Motherwell pair James McFadden and Stephen McManus also see their international exiles extended.