Israeli President Reuven Rivlin, meanwhile, spoke with Mexican counterpart Enrique Pena Nieto and said he was "sorry" about the "misunderstanding".
Netanyahu wrote on Twitter that Israel "will continue to have good relations" with Mexico.
"I believe our ties are stronger than any passing disagreement or misunderstanding," he added.
Mexico demanded an apology after Netanyahu tweeted last week in support of Trump's proposed wall spanning the Mexican border, saying a similar plan along Israel's southern frontier had been a success.
Netanyahu did not apologise, saying he did not comment on US-Mexico relations in the original tweet.
But he said he had a "long, fruitful and very friendly relationship" with the Mexican president.
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Israel's Rivlin spoke with Pena Nieto on Tuesday, as the two countries sought to calm the row.
"I am sure that nobody intended to compare between the situation of Israel, and the situation of Mexico," the Israeli president's office reported Rivlin as saying.
"The ties between us are so very strong and important, and we must leave behind us any such misunderstanding," he said.
"I am sorry for any hurt caused as a result of this misunderstanding."
Mexico's foreign ministry said that Rivlin had apologised and that Pena Nieto told him that the "tweet confounded Mexico and the Mexican Jewish community, and hurt relations between the two countries".
The Mexican leader said his country "has the desire to maintain mutual friendship and cooperation" with Israel.
Pena Nieto cancelled a planned meeting with Trump over the proposed wall.