I think I understand what you are looking for. Is the button on the Opportunity Detail Page?

I made few changes to the Controller and visualforce page. 


The controller has an action method sendEmail that is called from Visualforce Page's action. So as soon as page load it calls the method and redirects to the gmail. Basically you dont need getters anymore, but I left them (and calling from constructor). in case you are using for something else. 

public class MailOppController{

    public List<OpportunityContactRole>cRole{get; set;}
    public List<OpportunityContactRole>pRole{get; set;}
    public String oppID;
    
    private String primary;
	private String carbonCopy;
    
    public MailOppController(ApexPages.StandardController controller){

        Opportunity currentobject = (Opportunity) controller.getRecord();
        oppID = currentobject.Id;
        
        cRole = [SELECT ContactId, OpportunityId, IsPrimary FROM OpportunityContactRole WHERE OpportunityId = :oppID AND IsPrimary=false];
		pRole = [SELECT ContactId, OpportunityId, IsPrimary FROM OpportunityContactRole WHERE OpportunityId = :oppID AND IsPrimary=true];
    	
        getCarbonCopy();
        getPrimary();
    
    }
    
    public Pagereference sendEmail(){
        
        Pagereference emailPage;
        
        emailPage = new Pagereference('https://mail.google.com/mail/u/0/?view=cm&fs=1&tf=1&source=mailto&to='+primary+'&cc='+carbonCopy);
        
        return emailPage;
    }
    
    public String getCarbonCopy(){
        
        for (Integer i=0; i<crole.size();i++) {
            
            Contact mail_temp = [SELECT Email FROM Contact WHERE Id = :cRole.get(i).ContactId];
            
            if(i==0){
                carbonCopy = mail_temp.Email;
            } else {
            	carbonCopy = carbonCopy +', '+ mail_temp.Email;
            }
    	}
        return carbonCopy;
    }
    
    public String getPrimary(){
        
       
        
		Contact mail_temp = [SELECT Email FROM Contact WHERE Id = :pRole.get(0).ContactId];
        primary = mail_temp.Email;
        
        return primary;
    }
}

The visualforce page is much simplified now with just one <apex:page> tag

<apex:page standardController="Opportunity" extensions="MailOppController" tabStyle="Opportunity" action="{!sendEmail}"> 
    <!-- You dont need this anymore
    <apex:outputLink value="https://mail.google.com/mail/u/0/?view=cm&fs=1&tf=1&source=mailto&to={!primary}&cc={!carbonCopy}" target="_blank" id="theLink">GMail</apex:outputLink>   
    -->
</apex:page>

When you add this as standard button, please make sure you choose open in new window property.


Hope this helps.

Thank you