The most common defense I've heard in Renner's favor – and the one that he himself seems to use – is "she's just a fictional character, you people are too sensitive".
Yes, it's true that Black Widow is just a fictional character. But this doesn't negate the implications of calling her a slut.
Fictional characters are not real people – I'm sure that most fans would agree. But it is undebatable that fictional characters can and will inspire real people, maybe even more so than actual real people.
Consider this: have you stood up against authority because that's what Katniss did? Or stopped yourself from hurting a friend because you would never stoop to Peter Pettigrew's level?
In a society where the influence of pop culture is becoming more real and more tangible, fictional characters and how we see them have a real impact on how we, as real people, act and think.
This precisely lies the point that Renner doesn't understand – the implications of calling Black Widow a slut.
First, let's see WHY he's calling he a slut:
"I was asked the question like, so Black Widow’s been linked to Hawkeye, Iron Man, Bruce Banner, and Captain America, so what do you think about that. Well, I said, ‘It sounds like she’s a slut.’ Now, mind you, I was talking about a fictional character and fictional behavior, but Conan if you slept with four of the six Avengers, no matter how much fun you had, you’d be a slut."
So sleeping/being in a relationship/flirting with multiple members of the opposite sex makes a woman a slut.
How about James Bond? He's been in sexual relationships with dozens of women (76 according to the James Bond wiki). 76, compared to Black Widow's 4. But he's never been called a slut or man-whore; his sexual promiscuity has never been called into question.
Problem 1 – the double standards for men and women.
Then there is the implication that no matter what women do, they will always be judged by who they sleep with and who they are sexually or romantically involved with.
I doubt that there's anyone who would argue that Black Widow isn't a powerful, successful, intelligent woman, who is less valuable as an Avenger than her male teammates. But who she is is still defined as a slut.
Problem two – no matter how capable or brilliant you are, you, as a woman, is still no more than who you sleep with.
This links to the issue of how this affects real people and their behaviors and values. Many young girls look up to Black Widow as a role model, and while she is generally portrayed as a strong, independent woman (debatable as of AoU, but that's a post for another day), she is judged by men to be a slut. What does this tell young girls? That they will never be as valued as men? That they will always be sexualized? Or that the moment they begin a romantic relationship their worth as a human being is annihilated?
Is that what you want your friends, sisters, daughters to believe?
THIS is precisely what's wrong with Renner's accusations of Black Widow as a slut. The fact that she's a fictional character doesn't render the implications of his accusations null. In fact, it makes them even more powerful.