Almost, pRth/prath means to extend, increase, become famous, etc., and pRthvI/pRthivI, feminine of pRthu, literally spread, does mean earth, the entire world. pRtha is a spread palm (of hand), an unit of measure; and pRthA, the feminine of that, is the name of the daughter of the yAdava shUra/vasu who was adopted daughter of kunti/kuntibhoja, and was therefore also called kuntI, daughter of kunti. To the best of my knowledge pRthA was not used synonymously with pRthvI.
pArtha does mean son of pRthA. It could also mean son of pRthi, or short of pArthiva, a prince, or short of pArthona, the Sanskritized transliteration of the Greek name for the zodiac sign Virgo; but the famous usage was as son of pRthA.