Israel is the reason why Palestine isn't universally recognized
De facto states or quasi states that are not universally recognized are in this position because one of their neighbour has a strong political or historical objection to their existence/independence, and this neighbour's allies back it up:
Cyprus, Greece, EU oppose the recognition of Northern Cyprus' statehood
Morocco opposes the recognition of Western Sahara's statehood
China opposes the recognition of Taiwan (and Taiwan hasn't actually asked for separated statehood)
Serbia opposes the recognition of Kosovo's statehood
Israel opposes the recognition of Palestine's statehood
(let's not debate the motivation or the validity of each country's claim)
A two-state solution necessarily includes in recognition of the Palestinian state by Israel.
If a peace process ever results in a two states solution, it implies that Israel and Palestine will recognize each other's statehood and right to exist. They should then start diplomatic relationships that may be more or less cordial but exclude the state of war.
Because, tautologically, peace is what a successful peace process achieves.
Other countries should follow
Once Israel recognizes the State of Palestine, none of its allies in Northern America or Europe will have any reason left not to so as well. That's why it is very much expected that if a peace process results in a two states solution, Palestine will be universally recognized and become a full member of the United Nations.
A past example is East Timor, which was fighting for its independence from Indonesia between 1975 and 1999. Many countries, for instance Australia and the USA, didn't recognize East Timor's statehood because the Indonesian ally was far too important:
A staunchly anti-communist Indonesia was considered by the United States to be an essential counterweight, and friendly relations with the Indonesian government were considered more important than a decolonisation process in East Timor.
However, when Indonesian finally accepted East Timor's independence on May 20, 2002, no less than 27 States, including the USA and Australia, immediately recognized East Timor, and many others followed.