It is certainly possible to use general relationships to predict human population density or abundance. The relationship between population density/abundance and body size is an old topic in ecology that falls within the field of allometrics (how different features of organisms scale with body size). A closely related allometric relationship is Kleiber's law, which deals with the relationship between body size and metabolic rate.
Your assumption of a generally negative relationship between body size and abundance is correct, and the relationship is approximately linear on a log-log scale. As mentioned in the comments, the exact shape of the regression will depend on what organism groups are included, since the scaling relationship is known to differ between taxa. However, to get a reasonable ballpark figure we can look at a the classic paper by Damuth (1981), which include 307 mammal herbivores. The scaling relationship found there is:
$$D = 10^{4.23}*W^{-0.75}$$
where density (D) is in individuals per km2 and body weight is adult body mass in grams. Using your example of 70kg for humans would result in a predicted density of 3.9 individuals/km2. The scaling coefficient -0.75 (or -3/4) is often labelled "Damuth's rule", and is usually also found when pooling both ectotherms and endotherms.
However, remember that this regression was based on primary consumers, i.e. mainly herbivore grazers. Alternative data for groups of mammals can be found in Peters & Raelson (1984). That paper presents regressions for both carnivores+omnivores as well as for all groups of mammals (herbivores, carnivores, omnivores). The relationship for carnivores + omnivores (CO) is:
$$D = 15.7*W^{-1.15}$$
and the regression with all groups is (HCO):
$$D = 40.7*W^{-0.859}$$
In both of these regressions body weight is expressed in kg. Using the CO regression results in a predicted human density of 0.1 individuals/km2, and the HCO regression gives a human density of 1.1 individuals/km2.
Looking at these values, it is reasonable to assume that the predicted "natural" population density of humans lies in the range 0.1 - 4 individuals/km2, but certainly biased towards the lower part of the range. I suspect that a regression looking only at primates would predict an even lower value, so I would not rule out values below 0.1 individuals/km2.
To get a grasp of what this means in actual population numbers, we can extrapolate the densities to the entire earth. If we use the predicted density from the carnivore+omnivore model in Peters & Raelson (1984) and the entire land area of the earth, which is ~150 million km2 (Wikipedia: Earth), the predicted human population would be about 18 million. Using the high estimate from Damuth (1981) would correspond to about 600 million people. However, remember that this is using the entire land area, including deserts, mountain ranges and arctic areas.
Another point of reference is the estimated historical human population density during the Paleolithic, when we lived as hunter-gatherers. I have often seen population densities in the area of 1 per mile2, i.e. 2.6 individuals per km2. These estimates are naturally very uncertain, but shows that the allometric scaling relationships seem to provide at least reasonable order-of-magnitude estimates of "natural" human population density.
It is also somewhat enlightening to consider that our current human population size of 7 billion corresponds to the predicted population size of a 380 gram carnivorous mammal (using the CO relationship from Peters & Raelson (1984)), making the unrealistic assumption that this mammal has managed to colonize all land areas on earth. This shows pretty clearly what an outlier modern humans represent.
The generality of these types of relationships for predicting densities of individual species can however be questioned. For instance, the regression relationships also differ between climatic regions (see Peters & Raelson (1984) for more), and, as indicated in the comment by @remi.b, it is also unclear what population states that should be included. For instance, should only population densities in "ideal" habitats be used, when we know that population densities of the same species can differ by orders of magnitude in different habitats? It is also unclear if/how to consider the effects of modern humans. Silva & Downing (1995) also argue that the scaling relationship is nonlinear on the log scale, with a more shallow slope for larger species, and also find essentially no relationship between body size and density within some groups of species. More recent papers have also highlighted that studies on global relationships between density and body size often mix-up several different processes and relationships, see White et al (2007) for an overview on this.
