No state in the world needs a "legal right to exist." States exist if they fulfill the requirements of statehood. These requirements are the four requirements generally described as the Montevideo formulation: a permanent population, sovereign territory, an effective government and the capacity to carry on foreign relations. Israel fulfills these requirements and it is therefore a state. Some international legal scholars insist there is a fifth requirement — legal recognition of statehood by other states, generally through acceptance in the UN. Israel is a member state of the UN and therefore fulfills this requirement as well, if it is necessary.
The General Assembly resolution recommending the "Partition Plan" (General Assembly resolution 181) demonstrates that the General Assembly supported the establishment of a Jewish state in at least part of the remaining territory of the British mandate of Palestine which had previously been designated for the creation of a Jewish homeland by the League of Nations. This is an interesting fact, but if the General Assembly had never passed the resolution, it would not in any way affect the legality of Israeli statehood.