Jews believe that failure to do circumcise will lead to karet, or excision from being one of Gods people, as stated in Genesis. Indeed, circumcision is so much a part of Judaism that grown men, if they convert to Judaism, must be circumcised.
The fallacy of relying on Genesis or Leviticus for guidance on the importance of circumcision is that so much of the Tanakh is ignored by modern Jews already. So why stick with circumcision? A few examples of ignored laws:
Leviticus 5:1-10 If you commit a sin then you can expurgate your sin by sacrificing a goat by gouging it through its neck. If you can't afford a goat then two turtle doves are okay.
Leviticus 20:13 Gay men must be stoned to death.
Leviticus 25:44 Slavery is perfectly acceptable;
Lev 25:46 Any slave you own is hereditary property that you can happily pass onto your children when you die;
Deut 21:1-29 If you find a dead human body on your land then you must decapitate a female calf and wash your hands in its blood over a swiftly running stream.
Deut 21:10:14 Rape is perfectly acceptable as long as it is an enemy woman you are raping.
Deut 21:18-21 If your son doesn't do as he says, then feel free to flog him. If he still doesn't do as you ask, the priests will organize some blokes to stone him to death on your behalf.
I could go on and on with the number of rules and commandments that have been abandoned (quite rightly too) by modern Judaism. If the Tanakh is the inerrant word of God, then all these rules must be kept. If the Tanakh is not the inerrant word of God, then why is circumcision so utterly important? Cannot circumcision be ignored just like most other archaic laws are ignored?
And indeed there is a growing movement of contemporary Jews who do not accept the Tanakh's requirement to circumcise. This movement has developed a ceremony called the Brit shalom , or Covenant of Peace that takes the place of the traditional Jewish Brit Milah or Covenant of Circumcision. The movement relies on Leviticus 19:28 (Do not make gashes in your skin) and Deuteronomy 14:1 (Do not mutilate yourself) for its religious basis. And thus it is clear that being Jewish doesn't mean that circumcision must be a requirement to maintain the faith.