The background checks portion of the bill will be defeated today. It was being added as an amendment to the bill, and under a consent agreement all amendments needed 60 votes; it's not going to get there.
1. The correct thing to say about this is that the amendment was defeated by filibuster. It's a little tricky, but that's the essence of it. The UC agreement under which the amendment (and all amendments to this bill) is considered requires 60 votes; that's agreed to in order to avoid a cloture vote, which is necessary because of the standing GOP plan to filibuster everything. It is wrong to say that "Senate procedure requires 60 votes." Senate procedure only requires 60 votes in case of filibuster. Which, under current conditions, means in virtually all cases. Manchin-Toomey is defeated by filibuster.
2. All that said: anything that gets only four Republicans wasn't going to win in the House. Right now the House is split 232-201. Four Republican Senators is 9% of GOP Senators; the same percentage in the House would be 20 Republican votes, which means a tiny majority in the totally unrealistic case that every Democrat voted for the bill. So there's a very good chance there isn't even a simple majority in the House. Which wouldn't be close to enough: for Republicans to be willing to take up the bill at all, most mainstream conservatives would want the bill to come up (although not necessarily be willing to vote for it). That's simply not going to happen on a bill that gets 4 of 45 Republican Senators.
3. It's also important to remember that Manchin-Toomey is one of a series of possible provisions in an overall bill, and that filibuster rules also protect the majority to some extent. The 60 vote threshold that's blocking Manchin-Toomey may be necessary to prevent poison pill amendments from being adopted. Indeed, knowing that it takes 60 to get anything into the bill should change the opponents' strategy: instead of trying to design an amendment that would get a majority and then lead to the majority party abandoning the bill, the opponents may simply design amendments that will lose but supply votes which can be exploited later on.
4. Note that in the House, governed as it is by strict majority party rule, the majority party doesn't have to worry about popular amendments (or bills) that they don't want; they simply refuse to allow votes on them.
5. It is not entirely clear that a majority party rule system in which popular amendments and bills do not receive any vote at all, even if they would get a majority of the full chamber, is more democratic than a system in which everyone can force votes on any bill or amendment, but it takes 60 votes for passage. It's not even totally clear that a chamber in which every amendment and bill could always get a simple-majority vote is more democratic (because of the poison pill problem, and more generally because of unstable "majorities" on issues), but that may be irrelevant because in practice it's hard to get that middle ground.