Can Jewish people be spiritual but not religious, or does that go against their belief system?

Answer 1

Yes we as Jews can be spiritual and not be religious at the same time.
Most of the Orthodox Jews are spiritual and religious in their believe. I consider myself to be both spiritual and religious.
My cousin he is a Conservative Jew and he counts himself 50% spiritual and 50% religious and he is happy with that.
My other cousin she is a Reform Jew and she only counts herself as spiritual person who believes in God but not in all of the laws. However, since Judaism is a spiritual journey for her she chooses for her children to continue with her on that journey so she is married to a Jew who is also on a spiritual track of the religion.
A friend of my he is a Humanistic Jew and he believes both in the spiritual perspective of religion and on the mystical part of the religion. i.e. Humanistic approach mixed with some aspect of Kabala without Torah per se as a driving force nor the Talmud.
Let me just tell you this...there is no such thing as a bad Jew there are just those who except more of the religion including all of the laws, traditions, and spirituality that comes with that and those who intake less of the religion and more of spirituality.
I just pray that regardless of the spirituality that all Jews do three things safely within their believes 1) continue on with Jewish education 2) go to their type of synagogue every Shabbat 3) marry Jewish and raise children Jewish that way we all would have something to offer to everyone both in spiritual levels and religious levels.
I hope that this answers your question.

Answer 2

Anybody can be spiritual but not religious. In fact, the majority of Jews are secular.