When one spouse does not wish to practice birth control, but the other insists upon it, the Church allows the non-practicing spouse to engage in relations with the contracepting spouse for the sake of maintaining the marriage. The non-practicing partner is not allowed to use birth control himself, he must make his opposition to birth control known, and -- at appropriate times -- he should seek to educate his spouse about the morality of the practice. But, he can tolerate the fact that a contracepting spouse insists on birth control. In this case, you cannot agree to have a vasectomy, but you can engage in relations with your wife even if she insists on using birth control herself.
Frankly, I think you have larger issues here than birth control that you would do well to consider taking to a Catholic marital counselor: Your wife not only does not respect you (as demonstrated by your account of her emotional and verbal abuse of you), but she insists that you
must be the one to permanently sterilize yourself (and thus mutilate your body) on threat of divorce. These are serious issues that should be addressed as soon as possible. I recommend contacting the Pastoral Solutions Institute
for a referral to a Catholic marital counselor in your area. If your wife refuses to seek therapy with you, I urge you to consider going by yourself. God bless.
Vademecum for Confessors Concerning Some Aspects of the Morality of Conjugal Life
by the Pontifical Council for the Family