Assuming the ground note to be a C:

  Row, row, row your boat, gently down the stream
  C    C    C   D    E     E   D  E    F   G
  merrily, merrily, merrily, merrily, life is but a dream
  C' C'C'  G  G G   E  E E   C  C C   G    F  E   D C

The melody is very natural, using only straight harmonics: assuming C (1/1), we only need D (8/9), E (5/6), F (3/4),G (2/3) and C' (1/2), so the song can be played on the simplest of instruments such as a PVC pipe or a loose string.

It is also natural in that the strongest harmonics coincide with the strongest points of stress rhythmically. The strongest interval, C-C', divides the two parts of the song; the next strongest interval, C-G, divides each half; the next strongest interval, C-E-G, marks the next strongest notes in the melody; the remaining notes are 'fillers'. You can fill in the details of the melody by consecutively filling in the notes, going from the harmonically strongest to the harmonically weakest.

This is not accidental, but a consequence of the way melody fundamentally arises from harmony.

What I've always liked about this song is the 'row, row' bit. By repeating the ground note, it makes the 'row, row' resound through the rest of the song, even when it isn't actually sung. The act of rowing is repetitive in just this way. So the melody, as simple as it is, illustrates the lyrics quite cleverly.