The Pecos River in Texas, USA may arguably fit the description of a "saltwater river". A point of argument is what is considered to be "saltwater". For comparison, here are some bodies of water and their salinity.
* 35,000 ppm Pacific and Atlantic Ocean
* 13,000-23,000 ppm Black Sea
* 12,500 ppm Caspian Sea
* 10,000 ppm Baltic Sea
Note though, that salinity varies based on where, and at what depth you measure it. For example, the Baltic Sea has salinity as low as 5,000 to 8,000 ppm at the surface of the central basin.
At its saltiest, the Pecos River is reaching the lower end of the range above. Different sources give different threshold between the definitions of freshwater, brackish water, and saline water. Most would probably deem the Pecos as brackish (which would describe the Baltic Sea by the same standard), but it is salty enough to be a problem for human use. There are a number of sources that discuss the issue of the Pecos River and its salinity, but I'll direct you to this one, and quote a little from it
The Pecos River is among the saltiest rivers in North America with salinity levels regularly exceeding 7,000 ppm at the Texas and New Mexico border and 12,000 ppm near Girvin, Texas. High salinity in the river has adversely affected the stability and biodiversity of the riparian ecosystems, as well as, the economic uses of the river and reservoirs.
In general, natural sources of salt throughout the watershed cause the Pecos to be salty.
If you look at the slides in this presentation you'll see that different measurement locations range from around 1,000 ppm to 12,000 ppm (it gives measurements in mg/l, but that is roughly equivalent to ppm). It also gives historical measurement graphs for some select stations. The presentation also highlights that there is a history of efforts to engineer a reduction of the river's salinity.
The source of the Pecos River's saltiness is not so much surface salt as it is subsurface sources in the form of saline aquifers that feed the river through salt springs, for example. This research paper suggests that although the Pecos was naturally salty, the activities of man have increased the salinity. 1022
The major activity is irrigation, through two mechanisms. One is the diversion of water for irrigation, which allows increased evaporation in the fields, and irrigation water that returns to the river with more concentrated salt content. The other is the depletion of freshwater aquifers through pumping. The Pecos watershed overlays both saline and freshwater aquifers, both of which have historically contributed to the riverflow. Man has ignored the saline aquifers, but heavily pumped the freshwater aquifers, reducing their contribution to the river, but not the contribution of the saline aquifers.
This report is also good, and goes into much detail about the sources, both natural and human, of the Pecos River's salinity.
The upper reaches of the Red River in Texas overlay the same formation that creates saline conditions in the Pecos River. The streams and springs that form the source of the Red River are quite salty, and it is only further downstream, after dilution from other freshwater inputs that the Red River loses its high salinity. As this USGS fact sheet says:
Salinity is the greatest limitation on water use in the Red River Basin and is largely the result of naturally occurring salt springs in parts of the upper reaches of the basin (<PERSON> and others, 1988). The salt sources contribute water with large (relative to potable water) concentrations of dissolved solids, principally chloride. At certain times and locations, the salinity of streams in the basin exceeds that of seawater (<PERSON> and others, 1988).
Salty bed of Little Red River, a tributary of Red River, Texas (Wikipedia)
There are salty rivers in various places on Earth, but they are primarily found in arid regions. Since much of their flow comes from groundwater (which may be much saltier than rainfall), and evaporation from the river surface is high, it seems reasonable to me that they would be more likely to be saltier than rivers in wet regions that have large contributions from rainfall, and less net evaporation. These factors are magnified by human activities such as irrigation which happens in arid regions, but less so in areas of ample rainfal. 1022