Word for awkward situation

Solution 1:

David put his foot in his mouth.

foot in one's mouth, put one's: Say something foolish, embarrassing, or tactless. For example, Jane put her foot in her mouth when she called him by her first husband's name. (http://www.dictionary.com/browse/foot-in-one-s-mouth--put-one-s)

The mistake itself could be called a blooper. If you want something more old-fashioned: an indiscretion.

If David wants to express regret for what he said, he could say, "Me and my big mouth."

Solution 2:

Awkward IS the word.

"David says to Jane: "Wow, that John is a really boring guy to hang out with."

(...at that moment, John walks into the room, accidentally hearing the conversation between David and Jane.)"

Jane looks at David and whispers - Awkward !

Solution 3:

1. shoot off (one's) mouth/face
   Slang

   1. To speak indiscreetly.

   2. To brag; boast.

      -- Freedictionary

      "The whole damned bunch will fall apart ; all because you have to go and shoot your mouth off — you cock-eyed, mush-faced, cake-eating son of a so- and-so !"
      -- Hollywood Holiday: An Extravagant Comedy, Benn Wolfe Levy, John Van Druten, 1931

2. faux pas
   noun [ C ] US ​ /ˈfoʊ ˈpɑ/ plural faux pas /ˈfoʊ ˈpɑ, ˈpɑz/ ​
   a remark or action in a social situation that is a mistake and causes embarrassment or offense:

   She made the faux pas of referring to Wales as “part of England.”

   -- Cambridge Dictionary