Men may die by suicide more often, but it can’t be said that they’re all (or even mostly) dying by suicide because of domestic violence. Suicide and domestic violence are essentially separate issues which sometimes overlap. To insinuate that men’s greater suicide risk is attributable to domestic violence is an unjustified leap of logic. That said, women experience mental and emotional health problems due to abuse as well — including, sometimes, suicide.

Regarding your question about why I say that abuse poses a greater public health threat for women, here are some quick stats. (And I’ll link to a couple of sources at the bottom where you can find this info, rather than linking to each point — more expedient, but all the facts are there. ;) )

• Women are more likely than men to be murdered in a domestic violence situation.
• Additionally, very close to 100% of victims in domestic violence murder–suicides are female.
• Men are more likely than women to be physically violent with their partners.
• Over twice as many men as women “experience severe intimate partner violence” — and over 3 times as many women (compared to men) have been injured by a partner.
• Rape is a form of violence sometimes seen in abusive relationships, and women are more likely to be raped than men.
• In cases of rape, women are at greater risk than most men for contracting STIs, given usual anatomical differences and the fact that women are often asymptomatic.
• People with uteruses are also exclusively at risk of pregnancies (and, accordingly, of all the health issues and risks that go along with pregnancy and childbirth). As most men do not have a uterus, this, also, leaves women in abusive relationships facing a higher health risk than men.

As promised, the sources:

https://ncadv.org/statistics

https://www.womensaid.org.uk/information-support/what-is-domestic-abuse/domestic-abuse-is-a-gendered-crime/

https://www.cdc.gov/std/health-disparities/stds-women-042011.pdf

Nobody’s saying that abuse doesn’t happen to men or isn’t bad for men. But all said, if you’re considering risks like grave physical injury, homicide, rape, and the health-deleterious consequences of rape (e.g., infection and complications of pregnancy/birth), then women are indisputably at higher risk, and abuse poses a larger health threat to women.

You don’t have to believe me, and I’m not interested in spending time debating this further. But hopefully you find the links useful!