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If a king is not in check, can it capture an opposing chess piece?

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Yes, it can, as long as the opposing piece is not protected. It's even possible when the king is in check; all that matters is the position after the piece is captured. If there is an opposing piece attacking the king, the move is illegal; if not, it's legal.

[FEN "kR6/2B5/8/8/8/8/6r1/6nK b - - 0 1"]
[startflipped ""]

1... Ka7 {This is Black's only legal move.} 2. Kxg2 {However, this is a legal White move.}

Black to move can't capture the rook, since it's protected by the bishop. White to move (disregarding the situation in the upper left corner) can capture the rook, since it's not protected, but not the knight, since the rook protects it.

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    Nitpick - the question is "If a king is not in check". So maybe change the white rook to a knight? – Ian Bush Jan 3 at 9:26
  • This answer needs to mention that a king can capture an opposing piece even if it is in check (but won't be after the capture). – vsz Jan 3 at 11:06
  • @vsz thanks, I had a 'not' too many there. – Glorfindel Jan 3 at 13:32
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If the king is not in check, or even if it is in check, it can capture an opposing piece as long as so doing would not put or leave it in check, which would be illegal. The king can be a strong attacking piece, particularly in the diminished force endgame when it would be less likely to be putting itself in jeopardy.

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