How many games must be played in each of these league to recognize
that a player has completed a rookie season?

NHL

Rookie Qualifications
To be considered a rookie, a player must not have played in 25 or more
NHL games in any preceding seasons, nor in six or more NHL games in
each of any two preceding seasons. Any player at least 26 years of age
(by September 15th of that season) is not considered a rookie.
http://www.nhl.com/nhlhq/faq/hockey_ops.html#rookie

MLB

Qualifications for a Rookie: not more than 45 days on a major league
roster from opening day to August 31, inclusive, and had not exceeded
90 official at-bats or 45 innings pitched.
Source: 2003 New York Yankees Information & Record Guide
http://www.yesnetwork.com/announcers/article.asp?article_id=114

NFL

According to the NFL Players Association Contract:
"... a?rookie? shall be defined as a person who has never signed an
NFL Player Contract..."

http://64.233.167.104/search?q=cache:l1zMWEYFtuQJ:www.nflpa.org/PDFs/Agents/AgentApplicationSraSample.pdf+qualifications+rookie+%22never+signed%22+nfl&hl=en

I have not found anything concerning the NBA but I would suspect that
the rules are similiar to the NFL.  I think the MLB and NHL have
"minor leagues" or "farm system" so players are called up and sent
down more often.  That is why they have such specific rules.