The Register of Debates (which covers Davy Crockett's first two terms in Congress from 1827-1831) and the Congressional Globe (which covers his last term in Congress from 1833-1835) do not provide verbatim transcripts of speeches made on the House floor. There is no way to know therefore if the text of the speech that has been widely distributed is true to what Davy Crockett said.
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However, there is a historical record that supports a similar story - the House considered a bill of relief for the family of deceased general Brown in April of 1828 and Davy Crockett is on record opposing that bill and offering personal support to the family. You can read the (very brief) summary of that in the Register of Debates here. Crockett's comments are summarized at the bottom right of the page.