There is a National Highway System in the US, which identifies priority roads for federal investment. The NHS includes approximately 160,000 miles of roadway— principally the interstate highway system—as well as other roads that are important to the nation’s economy, defense, and mobility.
Should there be a similar National Transit System? The American Public Transportation Association has advocated in its strategic plan to "Work to define elements of a national transit system that complement the existing national highway system, including those elements of public transportation systems that are of national and regional significance.", but otherwise there seems to be no national policy discussion of this.
This of course has a lot to do with history. The appropriate federal role in transit, even in interstate passenger service, remains a long standing question. Danielson (1965), discussing the funding crisis facing the then private commuter railroads in the late 1950s and early 1960s, writes "Support from outside the government for [the position that there was no federal role for mass transportation] came in 1959, when, in an authoritative study on railroad policies for the influential Brookings Institution, James Nelson concluded that support of [commuter trains] was a local matter."
President-elect Kennedy supported a $100 million mass transportation loan program proposed by New Jersey Senator Harrison Williams. This eventually passed as a $50 million loan, and $25 million demonstration-grants (not for long-term capital improvements), as well as a small planning program, but Danielson (p.158) writes "While loans might be more palatable to Congress, Budget officials pointed out that they would offer little incentive to communities which had reached their debt limit or which could float municipal tax-exempt bonds at a lower interest rate than federal loans. They also emphasized that the only criteria in the Williams bill for setting the magnitude of federal financial participation had been how much Congress might be persuaded to accept." Loans were nevertheless endorsed by the urban coalition of metropolitan areas and the private railroads. Authorization was subsequently cut from $75 million to $42.5 million by the House, which was less sensitive to urban issues at the time than the Senate.
Mass transit was then housed in the Housing and Home Finance Agency (later the Department of Housing and Urban Development). This positioning of transit and housing together is echoed today by the Obama Administration's Livability Initiatives cutting across DOT and HUD. The concern of locating mass transit within the Department of Commerce was the influence of the Bureau of Public Roads.
In 1964, a $375 million urban transportation program was passed and signed by President Johnson. However this program took the form of grants rather than loans, which were by their very nature more favored by states and metropolitan areas.
In the post-Interstate era, there has been less new construction, highway expansion has slowed. More federal funds go to transit (about 25 percent of federal surface transportation capital funds go to transit), and the match requirement for states has increased.
Metropolitan areas are not generally connected by urban public transit in the US, though they are connected by Amtrak and by the Aviation sector (which is certainly mass transit, if not public transit), both of which are already seen as national systems. There are other aspects. Highways are infrastructure, but not carriers. Transit services are carriers, and sometimes infrastructure and carriers. Occasionally the services are interstate (in multi-state metros), but none are really national.
There is the issue of equity. People who live in urban areas are more likely to take transit (and less likely to take highway). If funds for transportation came from general revenue, there would be a strong equity case for treating the modes equally (and thus for a federally supported NTS). However, since most funds at the federal level come from highway user fees, it has less salience as an argument, since if urban areas have less highway travel per capita, they have less highway needs per capita and they generate fewer user fees per capita. The rationale for spending highway user fees on transit is in the end quite weak. ...
 Danielson, M.N. (1965) Federal-metropolitan politics and the commuter crisis. Columbia University Press. P. 40;
Nelson, J.C. (1959) Railroad transportation and public policy. The Brookings Institution.