And so, as promised, I'll do it.
Let's work with 1m long strips of gutter. It'll be easier to calculate everything starting from here.
Let's say the gutter is already full of ice. (We'll work on the ice filling problem later on)
The standard gutter size (according to this site) is 5-inch K-Style, or 6-inch half round. If we use the half round version, we can learn it holds around 9L of ice. The latent heat of fusion of 9L of ice is 3000kJ, or 833Wh. This means that if you wanted to melt this ice in an hour, you'll need 833 Watts of power. For each meter of gutter.
But what's the available solar energy?
Second assumption: Let's say the sun shines all day, with the perfect orientation regarding the gutter, and the gutter absorb all the energy it receives from the sun. Let's assume it's the shortest day of the year, at the US mean latitude (around 38°N).
According to the third chart, the length of day on the shortest day of the year is around 9h30m. Let's round this up to 10h (I like round and easy numbers for my ballpark calculations).
The cross section of our gutter is around 0.15m². The sun irradiance reaching the ground stands at around 1000W/m². This means around 150W reaches our gutter. On a 10h day course, that would be 1500Wh. Hey, that would be enough! Well, yes, if you take your gutter and put in the perfect orientation regarding the sun. Which it's not true.
In this case, the energy received will be lower. Moreover, one also has to take into account the efficiency of the energy conversion. High quality solar thermal collector (which collect solar radiation and convert it to heat) typically have efficiency at around 60%. This means, in our case (where the efficiency will be lower, we'll receive at best 900Wh.
If we take into account the fact that our gutter has a fixed orientation, the energy received will be even lower than that. Thus, we won't have enough energy to melt the ice.
Given this data, I'd say it's not possible.
As for the ice filling the gutter. The problem is still the same. Making sure that the water flowing from the roof stays hot enough will still means providing enough energy to keep it above freezing. Also, usually, water melts on the roof, but doesn't flow alone, i.e. it brings down some snow and ice in the gutter, which you'll have to melt of you want to prevent ice buildup in the gutter.