There is a process for permanent residents of Israel to become citizens. It is a long and drawn-out process, potentially taking years, much longer than the process of citizenship by immigration for Jews or those with Jewish ancestry.
There is also a process for refugees (including those from Palestine) to seek asylum in Israel and thus gain permanent residency. They could then seek citizenship after meeting the requirements for that process. Approximately 21% of Israel's citizen population are Palestinian Arabs, whether they (or their parents/grandparents) had been incorporated into the new 1948 state or whether they had gained citizenship later on.1 They enjoy full rights of citizenship and are equal under the law to Jewish (and other non-Arab) citizens.2
Therefore, yes, in theory there is a mechanism for Palestinians, including Gazans, to become Israeli citizens, but as far as I can tell, Israel is making no effort to make this any easier of a process. In fact, as other answerers have pointed out, Israel's right wing (which is currently in power) is adamantly against establishing a right of return for Palestinians who lived in British Palestine before 1948. Therefore, in practice, it is extremely unlikely that a Palestinian from Gaza would be able to obtain Israeli citizenship under "normal" circumstances, and effectively impossible during wartime.
1This figure does not include Palestinians living in East Jerusalem since 1948, who are considered permanent residents but not citizens (unless they have separately obtained citizenship), nor does it include Palestinians living in the West Bank or Gaza, who have no legal status in Israel and live under Israeli martial law, Palestinian civil law, or the Hamas terrorist regime, depending on where exactly they live, all of which are oppressive and brutal to varying degrees.
2There are several Arab political parties within the Israeli legislature, and during the brief period a couple of years ago in which the political center took control of the government from Netanyahu, they were even a part of the governing coalition. There is also an Arab sitting on the Israeli Supreme Court. That being said, it is not the case that Israeli Arabs do not experience racism or discrimination. For instance, they are often ethnically profiled by police forces, and there is a clear income gap between Arab and Jewish families. However, the Israeli Arab experience is significantly different from the oppressive occupation faced by West Bank residents.