The law already stipulates that women who are eligible for the plans and want abortion coverage must purchase that provision separately, with their own funds. Pitts' bill would clarify the process and require, among other things, that insurance providers offer an abortion coverage-free plan.
Rep. Mike Pence, R-Ind., has introduced the "Title X Abortion Provider Prohibition Act" which would curtail federal family planning grants for groups like Planned Parenthood, which provides abortion services and receives more than $350 million in federal dollars each year.
"The largest abortion provider in America should not also be the largest recipient of federal funding under Title X," Pence said last month. The bill "will prevent any family planning funds under Title X from going to Planned Parenthood or other organizations that perform abortions. It will ensure that abortion providers are not being subsidized with federal tax dollars."
Democrats have vowed to fight the measures and accused Republicans of violating their promise to focus on jobs and the economy.
"They're focusing on extreme legislation that takes away women's ability to make their own reproductive life decisions and choices," said Schultz. "This is severe and dangerous legislation and the fact that the Republican leadership is making it a priority … shows that they are more interested in highjacking a woman's independence over her own body rather than focusing on our nation's No. 1 priority: jobs."
ABC News' Huma Khan contributed to this report.