What specific part of the US constitution or the Senate/House rules states that votes (on legislation or nominations) are decided by a simple majority of lawmakers present? (Disregarding rules about cloture.) Or is this simply a matter of precedent?
The principle of simple majority vote is the assumed principle unless a rule specifies otherwise. Although the Constitution never indicates that a vote in Congress has to be taken by majority vote, simple majority is assumed. On certain actions, the Constitution provides for a vote to be taken by two-thirds vote or similar; however, all such language only considers the requirement to be two-thirds of the members present and voting.
There is a CRS report on Senate Quorum and voting. On page 11, you will find the generally accepted mechanism that simple majority vote is sufficient to pass almost all measures.
The American system began from the founders considering the past systems of governance and deciding the best form for the United States. Certain things were considered fundamental, such a majority rule.
In the early days of the founding of the USA, travel was slow. Congressmen elected in certain parts of the country might arrive late to Congress, but Congress may have had business to do. The business of the country does not stop when a few Representatives are missing. Thus the idea that not all members need to vote or be present for a certain issue was adopted. In order to prevent the abuse of this system, quorum was defined in the constitution as majority. Thus at least more than half of the members of a chamber need to be present. If some of the present legislators are really unhappy with a certain action, those legislators could simply leave and the chamber would be without quorum. In such a situation, proponents of the measure would then need to look to find a majority of the total membership of the chamber that is willing to show up and vote. Once a majority is present, business can be conducted. On any motion, if more vote in favor of it then against it the motion carries.
Furthermore the issue of simple majority was addressed by the Supreme Court in United States v. Ballin (1892). The court provides a historic precedent argument in addition to others why Congress votes by simple majority. There is also a remark that although some states require absolute majority in their legislatures, such language is not present in the Constitution. To those who do not wish to read the case: a law was challenged on the fact that it was passed by a vote of 138 to 3 in the U.S. House of Representatives. At that time, according to the Apportionment of the House from 1891-1900, the size of the House was 330 (the act in question passed in May of 1890). Thus 138 is less than half of the number of Representatives. However, the Supreme court ruled as follows:
If we appeal to the journal of the House, we find that a majority of its members were present when the bill passed, a majority creating by the Constitution a quorum, with authority to act upon any measure; that the presence of that quorum was determined in accordance with a valid rule theretofore adopted by the House, and that of that quorum a majority voted in favor of the bill. It therefore legally passed the House...
Additionally, while there is a question of whether a rule to increase the voting threshold to pass a bill is constitutional, the referenced paper makes it clear that without any rule specifying the voting standard, an action can be taken by simple majority, which is anything more than 50% of the members (usually abbreviated as 50% plus 1) who vote on a particular motion. From the literature, there does not seem to be any disagreement on this issue.
For example, this means if 60 Senators are present in the Senate, then there is quorum. If on a particular motion only 5 Senators vote and 3 vote in favor and this motion requires majority for adoption, the motion carries because a majority of those voting agreed to the motion.
An example of when a simple majority may take action, without an absolute majority, in Congress, is available here. The action was taken by a vote of 50-46. Absolute majority would require at least 51 votes; however, less than that amount approved the action. Here is another vote, which approved an action 48-43.
Article 1, Section 5, Clause 1 of the U.S. Constitution:
Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a smaller number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each House may provide.
The Senate assumes that it is complying with the Constitution (that a quorum is present), unless and until a Senator calls for a qurorum, after which the roll will be called to determine the presence of a quorum. From here (quoting from Riddick's Senate Procedure):
The Senate operates on the presumption that a quorum is present at all times, under all circumstances, unless the question to the contrary is raised, or the absence of a quorum is officially shown, or until a point of no quorum is made even though a voice vote is taken and announced in the meantime.
Nominations used to require three-fifths of Senators present to invoke cloture, but precedents set in 2013 and again in 2017 caused nominations for pretty much all judicial posts to require only a simple majority in order for debate on the nomination to be closed.
The same simple majority standard tickles down to each committee in the House in order to set each committee's specific rules, but they can stipulate that as few as two members of the committee constitute a quorum in the committee rules in the cases of taking testimony or receiving evidence, and not less than one-third of the committee for conducting other business (except for a few, which are listed below). A committee cannot make recommendations to the full body without a simple majority. From The Rules of the House, page 18:
(h)(1) A measure or recommendation may not be reported by a committee unless a majority of the committee is actually present.
(2) Each committee may fix the number of its members to constitute a quorum for taking testimony and receiving evidence, which may not be less than two.
(3) Each committee (other than the Committee on Appropriations, the Committee on the Budget, and the Committee on Ways and Means) may fix the number of its members to constitute a quorum for taking any action other than one for which the presence of a majority of the committee is otherwise required, which may not be less than one-third of the members.