I have heard several different explanations from secondhand sources as to why the Lubavitcher Rebe zt''l didn't visit Israel. Did he ever explicitly state his reasons for not going?
In his own words (on video) here in Hebrew with English subtitles (the second half of the video). He states two reasons.
1) Lack of solid Halachic authority to return if he did go to visit, and decreased influence over American Jews and newly arriving Jewish immigrants to America if he stayed in Israel permanently.
2) Concern about sending a message to Jews in America that it is better to be in America, since they would say he regretted leaving America.
Here is a posting giving the reason that the Rebbe stated.
by Rabbi Naftali Silberberg
[It is important to note that since the Rebbe took leadership of the Chabad movement in 1951, he never took a day off, and never traveled outside of New York.]
There are several possible reasons for why the Rebbe never visited Israel; some of them were voiced by the Rebbe himself in private on various occasions.
Halachically it is forbidden to leave the Land of Israel (although there are some loop-holes to this prohibition).
If the Rebbe would have visited Israel he would have been compelled to stay there, something which he did not want to do for the following reasons:
The gravesite of his father-in-law, the Previous Lubavitcher Rebbe, is in New York City. It is the custom of Chabad Rebbes to go to the gravesite(s) of their predecessor(s) to intercede on behalf of all those who need blessings and prayers.
The majority of Jews live in the United States. Certainly it would have been to Rebbe's personal advantage to move to the land of Israel and enjoy its holiness, but the Rebbe, the selfless leader of world Jewry wasn't concerned with his own welfare and spiritual advancement, rather he chose to remain in the location where he could maximize his influence on his fellow Jews. As the old adage goes, "the captain is always the last one off the sinking ship!"
Incidentally, the Rebbe demanded this sacrifice from his Chassidim as well. The Rebbe reiterated on many occasions that someone who lives outside of the Land of Israel, but is in a position where he/she is a positive influence on others, helping them in studying Torah or doing Mitzvot, it is his obligation to stay in his community and continue with this holy work.
When someone asked the Baba Sali this question he answered that Mosheh Rabeinu was not able to enter Eretz Yisroel b/c Moshe is compared to the sun and Eretz Yisroel is like the moon and only Yehoshua who was like the moon was able to enter Eretz Yisroel.. he finished off saying "it is enough for a wise person with a hint".....