"Can a white American never be suspicious of a black American?"
A white American can be suspicious of a black American but it should be with an awareness of the biases that color our interpretation of POC (people of color):
The differences that the dominant culture observes in anyone that doesn't fully fit in are seen as "not normal, primitive, uneducated," or some other descriptor that ultimately means "bad." This is a process called "othering." This is why it's bad to wear a hoodie.
Everyone, including POC and people that advocate against prejudice act in prejudiced ways subconsciously. This is called implicit bias  and it's echoed in our actions.
POC are constantly and unjustly profiled, harrassed and sometimes murdered by police. 
Without this awareness any behavior that the dominant culture is unfamiliar with can potentially be justification for "suspicion."
Even behavior that the dominant culture partakes in like "looking around," "walking in the rain at night" and "not walking along the sidewalk" can become suspicious.
Martin's behavior before Zimmerman saw him is irrelevant as to whether he deserved suspicion. It's also irrelevant that Zimmerman didn't know or recognize him. If Zimmerman was able to "think" Trayvon was Black he was capable of unconsciously acting out prejudice.
It is impossible to know whether or not someone was thinking racist thoughts either consciously or subconsciously but I still think that unless we know George Zimmerman was aware of and acted against issues like implicit bias we should assume he was biased. In my opinion it's our refusal to admit our implicit bias (due to our shame) that perpetuates racism.
You may have a problem with this idea that some characterize as "Guilty until proven innocent." Here is my response to that:
We all know that racism is strongly alive in America. No one denies that. POC live it though, white people don't. So when POC assume that white people are likely to be racist, or are acting in racist ways (like George Zimmerman), it's not shameful, it's accurate and it's a perspective they have to have in order to protect themselves from the racism that white people don't want to talk about.
When white people have a national conversation on the problems of a normative culture's judging and othering of other cultures and when white people have a conversation on implicit bias and all of its manifestations and when white people in majority can admit that they have implicit bias, maybe then it will start to become safe for POC to stop assuming that white people act in racist ways. Until then myself and POC will assume that when people act in racist ways, it's due to racism.
edit: I have a problem with my assumption that POC all have the same perspective as I do. That is a mistake and I will correct it.